Product Systems over Ore Monoids

Suliman Albandik and Ralf Meyer

Received: August 22, 2015 Revised: October 10, 2015

Communicated by Joachim Cuntz

Abstract. We interpret the Cuntz–Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz–Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz–Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.

2010 Mathematics Subject Classification: 46L55, 22A22 Keywords and Phrases: Crossed product; product system; Ore conditions; Cuntz–Pimsner algebra; correspondence; groupoid model; higher-rank graph algebra; topological graph algebra.

1. Introduction

Let A and B be C^{∗}-algebras. A correspondence from A to B is a Hilbert B-module E with a nondegenerate ^{∗}-homomorphism from A to the C^{∗}-algebra of adjointable operators on E. It is called proper if the left A-action is by compact operators, A → 𝕂(E). If E_{AB} and E_{BC} are correspondences from A to B and from B to C, respectively, then E_{AB} ⊗_{B} E_{BC} is a correspondence from A to C. A triangle of correspondences consists of three C^{∗}-algebras A, B, C, correspondences E_{AB} , E_{AC} and E_{BC} between them, and an isomorphism of correspondences u: E_{AB} ⊗_{B} E_{BC} → E_{AC}; that is, u is a unitary operator of Hilbert C-modules that also intertwines the left A-module structures. Such triangles appear naturally if we study the correspondence bicategory of C^{∗}-algebras introduced in [12]. This article started with the observation that a correspondence triangle with A = B and E_{BC} = E_{AC} is the same as a Cuntz–Pimsner covariant representation of the correspondence E := E_{AB} by adjointable operators on F := E_{BC} = E_{AC}, provided E_{AB} is proper. Thus we get to the Cuntz–Pimsner algebra directly, without going through the Cuntz–Toeplitz algebra.

This is limited, however, to proper correspondences and the absolute Cuntz– Pimsner algebra; that is, we cannot treat the relative Cuntz–Pimsner algebras introduced by Muhly and Solel [59] and Katsura [42]. The relative versions are most relevant if the left action map A → 𝕂(E) is not faithful. Then the map from A to the Cuntz–Pimsner algebra is not faithful, and the latter may even be zero. Our observation about the Cuntz–Pimsner algebra of a single proper correspondence has great conceptional value because it exhibits these (absolute) Cuntz– Pimsner algebras as a special case of a general construction, namely, colimits in the correspondence bicategory, see [2]. Other examples of such colimits are crossed products for group and crossed module actions, inductive limits for chains of ^{∗}-homomorphisms, and Cuntz–Pimsner algebras for proper essential product systems. In this article, we apply our observation on the Cuntz–Pimsner covariance condition to the case of Cuntz–Pimsner algebras for proper essential product systems over monoids. Much less is known about their structure. Following Fowler [33], they are always defined and treated through the corresponding Cuntz–Toeplitz algebra. Many articles never get farther than the Nica–Toeplitz algebra. We shall prove strong results about the structure of Cuntz–Pimsner algebras of proper essential product systems over Ore monoids. Commutative monoids and groups are Ore. So are extensions of commutative monoids by groups such as the monoid 𝕄_{n}(ℤ)^{×} of integer matrices with non-zero determinant, with multiplication as group structure: this is an extension of the group Gl_{n}(ℤ) by the commutative monoid (ℕ_{≥1} , ·). Thus most of the semigroups currently being treated in the operator algebras literature are Ore monoids. The main exception are free monoids, which are not Ore. Let P be a cancellative Ore monoid and let G be its group completion. Let A be a C^{∗}-algebra and let (E_{g})_{p∈P} be a proper, essential product system over P with unit fibre E_{1} = A; that is, the left action of A on each E_{p} for p ∈ P is by a nondegenerate ^{∗}-homomorphism A → 𝕂(E_{p}). The Ore conditions for P ensure that the diagram formed by the C^{∗}-algebras 𝕂(E_{p}) for p ∈ P with the maps

𝕂(E_{p}) → 𝕂(E_{p} ⊗_{A} E_{q}) ≅ 𝕂(E_{pq})

for q, p ∈ P is indexed by a directed set. Hence the colimit for this diagram behaves like an inductive limit; it may indeed be rewritten as an inductive limit of a chain of maps 𝕂(E_{p_{i}} ) → 𝕂(E_{p_{i+1}} ) for a suitable map ℕ → P if P is countable. Let O_{1} be the inductive limit of this diagram of C^{∗}-algebras. We construct a Fell bundle (O_{g})_{g∈G} over G with O_{1} as its unit fibre, such that its section algebra is the Cuntz–Pimsner algebra O of the given product system. Thus the construction of the Cuntz–Pimsner algebra of a product system over P has two steps: inductive limits and Fell bundle section algebras. After putting this article on the arxiv, we learnt of the preprint [49] by Kwaśniewski and Szymański, which proves essentially the same result about the structure of Cuntz–Pimsner algebras over Ore monoids.

Assume now that the correspondences E_{p} are full as Hilbert A-modules. Then the C^{∗}-algebras 𝕂(E_{p}) for p ∈ P are all Morita–Rieffel equivalent to A and the Fell bundle (O_{g})_{g∈G} is saturated. Let 𝕂 be the C^{∗}-algebra of compact operators. By the Brown–Green–Rieffel Theorem, E_{p} ⊗ 𝕂 ≅ A ⊗ 𝕂 as a Hilbert A ⊗ 𝕂-module, so we may replace the proper correspondence E_{p} by an endomorphism φ_{p}: A⊗𝕂 → A⊗𝕂. Choose a cofinal sequence (p_{i})_{i≥0} in P as above with p_{0} := 1, and let q_{i} ∈ P be such that p_{i} = p_{i−1} q_{i}. Then O_{1} ⊗ 𝕂 is the inductive limit of the inductive system

A ⊗ 𝕂 −−→^{φ_{q_{1}}} A ⊗ 𝕂 −−→^{φ_{q_{2}}} A ⊗ 𝕂 −−→^{φ_{q_{3}}} A ⊗ 𝕂 → ⋯;

this inductive limit carries a natural G-action with G ⋉ O_{1} ≅ O ⊗ 𝕂. Thus the K-theory of O_{1} is an inductive limit of copies of the K-theory of A; the maps are induced by the proper correspondences E_{q} or, equivalently, the endomorphisms φ_{q} of A⊗𝕂. Roughly speaking, we have reduced the problem of computing the K-theory for Cuntz–Pimsner algebras of proper product systems over an Ore monoid P to the problem of computing the K-theory for crossed products with the group G. This latter problem may be difficult, but is much studied. We cannot hope for more because crossed products for G-actions are special cases of Cuntz–Pimsner algebras over P. Lots of C^{∗}-algebras are or could be defined as Cuntz–Pimsner algebras of product systems over Ore monoids. Thus our structure theory for them has lots of potential applications. As a sample of how K-theory computations in this context might work, we consider certain higher-rank analogues of the Doplicher– Roberts algebras that motivated the introduction of graph algebras. (Our higher-rank analogues, however, need not be higher-rank graph algebras.) Many Cuntz–Pimsner algebras are constructed from generalised dynamical systems, such as higher-rank topological graphs. The appropriate topological analogue of a product system over P is given by locally compact spaces X and M_{p} for p ∈ P with continuous maps r_{p}, s_{p}: M_{p} → X and σ_{p,q}: M_{pq} ⟶^{∼} M_{p} ×_{s_{p} ,X,r_{q}} M_{q}. We assume r_{p} to be proper and s_{p} to be local homeomorphisms to turn (M_{p}, s_{p}, r_{p}) into proper correspondences over C_{0}(X). These form a product system over P with unit fibre C_{0}(X). The data above may be called a topological higher-rank graph over P; we prefer to call it an action of P on X by topological correspondences. In the above situation, we construct a groupoid model for the Cuntz–Pimsner algebra of our product system. This model is a Hausdorff, locally compact, étale groupoid. We translate what it means for this groupoid to be effective, locally contracting, or minimal into the original data (X, M_{p}, s_{p}, r_{p}, σ_{p,q}). We also describe invariant subsets and invariant measures for the object space of our groupoid model. This gives criteria when the Cuntz–Pimsner algebra of an action by topological correspondences is simple or purely infinite and often describes its traces and KMS-states for certain one-parameter groups of automorphisms. Our results are interesting already for the commutative Ore monoids (ℕ^{k}, +). Several authors have considered examples of product systems over these and other commutative cancellative monoids ([10, 31, 32, 51, 79]). Commutativity seems to be a red herring: what is relevant are Ore conditions. Commutativity is hidden also in Exel’s idea in [29] to extend a semigroup action to an “interaction semigroup.” Examples in [31] show that interaction semigroups for product systems over ℕ^{2} only exist under some commutativity assumptions about certain conditional expectations. Our approach shows that a deep study of these examples is possible without such technical commutativity assumptions (see Section 5.1). Topological higher-rank graphs are already very close to our situation, so we compare our constructions to the existing ones for this class as we go along. As an example involving noncommutative Ore monoids, we discuss how the semigroup C^{∗}-algebras of Xin Li [55] fit into our approach.

2. The Cuntz–Pimsner covariance condition

We first reinterpret the Cuntz–Pimsner covariance condition for a single correspondence as a nondegeneracy condition.

Definition 2.1. A correspondence from A to B is a Hilbert B-module F with a nondegenerate left action of A by adjointable operators. A correspondence is proper if A acts by compact operators. We often write the left action multiplicatively as a · ξ for a ∈ A, ξ ∈ F. An isomorphism between two correspondences from A to B is an A, B-bimodule map that is unitary for the B-valued inner products.

Definition 2.2. Let A and B be C^{∗}-algebras and let E be a correspondence from A to itself. A transformation from (A, E) to B is a correspondence F from A to B with an isomorphism of correspondences u: E ⊗_{B} F ⟶^{∼} F.

This definition is a special case of the standard notion of a “transformation” between two “morphisms” between two “bicategories” (see [52]). This point of view is developed further in [2]. Here it will not play any role besides guiding our choice of notation. We want to relate transformations to certain Toeplitz representations of correspondences. The next proposition is already implicit in [58, §5] and has also been used by other authors before.

Proposition 2.3. Let A, B_{1}, B_{2} be C^{∗}-algebras. Let E: A → B_{1}, F_{1} : B_{1} → B_{2} and F_{2} : A → B_{2} be correspondences. Isomorphisms E ⊗_{B_{1}} F_{1} → F_{2} of correspondences are in natural bijection with linear maps S: E → 𝔹(F_{1}, F_{2}) that satisfy

(1) S(aξ) = aS(ξ) for all a ∈ A, ξ ∈ E;

(2) S(ξ_{1})^{∗}S(ξ_{2}) = ⟨ξ_{1}, ξ_{2}⟩_{B_{1}} for all ξ_{1}, ξ_{2} ∈ E;
(3) S(E) · F_{1} spans a dense subspace of F_{2}.

Furthermore, (2) implies (4) S(ξb) = S(ξ)b for all b ∈ B_{1}.

Proof. First let u: E ⊗_{B_{1}} F_{1} → F_{2} be an isomorphism of correspondences. Define S(ξ)(η) := u(ξ ⊗ η) for ξ ∈ E, η ∈ F_{1}. For fixed ξ, this is an adjointable operator S(ξ): F_{1} → F_{2} because u and the operator F_{1} → E ⊗_{B_{1}} F_{1}, η ↦ ξ ⊗η, are adjointable. This map S clearly satisfies (1). Since u is isometric,

⟨η_{1}, S(ξ_{1})^{∗}S(ξ_{2})η_{2}⟩ = ⟨S(ξ_{1})η_{1}, S(ξ_{2})η_{2}⟩ = ⟨ξ_{1} ⊗ η_{1}, ξ_{2} ⊗ η_{2}⟩ = ⟨η_{1}, ⟨ξ_{1}, ξ_{2}⟩η_{2}⟩

for all ξ_{1}, ξ_{2} ∈ E, η_{1}, η_{2} ∈ F_{1}. This is equivalent to (2). Since u is unitary, it has dense range, which gives (3). Conversely, let S: E → 𝔹(F_{1}, F_{2}) be given. Define u on the algebraic tensor product of E and F_{1} by linear extension of u(ξ ⊗ η) := S(ξ)(η). Condition (2) ensures that this is an isometry and hence extends to the completion E ⊗_{B_{1}} F_{1}. Hence S satisfies S(ξb)(η) = S(ξ)(bη) for all ξ ∈ E, b ∈ B_{1}, η ∈ F_{1}, which is equivalent to (4). Condition (1) says that u is A-linear, and (3) says that it has dense range. Being isometric, this means that u is unitary. The two constructions u ↔ S are inverse to each other because u is determined by its values on the monomials ξ ⊗ η. □

A (Toeplitz) representation of E is usually defined as a map S satisfying (1) and (2) in the case F_{1} = F_{2}; we also allow the case F_{1} ≠ F_{2} for a while because this is used in [2] and in Proposition 2.8.

Definition 2.4. A representationn is nondegenerate if it satisfies (3).

By Proposition 2.3, a transformation from (A, E) to B is equivalent to a correspondence F: A → B with a nondegenerate representation of E by operators on F. We now relate nondegeneracy to the Cuntz–Pimsner covariance condition:

Proposition 2.5. Nondegenerate representations of a correspondence E are Cuntz–Pimsner covariant. The converse holds if E is proper.

Proof. Let A, B_{1} and B_{2} be C^{∗}-algebras and let E: A → B_{1}, F_{1} : B_{1} → B_{2} and F_{2} : A → B_{2} be correspondences as in Proposition 2.3. The left actions of A in our correspondences are nondegenerate ^{∗}-homomorphisms

φ_{E}: A → 𝔹(E), φ_{F_{2}} : A → 𝔹(F_{2}).

Let u: E ⊗_{B_{1}} F_{1} → F_{2} be an isomorphism of correspondences. The map

ϑ: 𝔹(E) → 𝔹(F_{2}), T ↦ u(T ⊗ 1)u^{∗},

is a strictly continuous, unital ^{∗}-homomorphism. It satisfies ϑ ∘ φ_{E} = φ_{F_{2}} because u intertwines the left actions of A. If ξ_{1}, ξ_{2} ∈ E and |ξ_{1}⟩⟨ξ_{2}| is the corresponding rank-one compact operator on E, then

ϑ(|ξ_{1}⟩⟨ξ_{2}|) = S(ξ_{1})S(ξ_{2})^{∗}.

This formula still defines a (possibly degenerate) ^{∗}-homomorphism ϑ: 𝕂(E) → 𝔹(F_{2}) for any representation S: E → 𝔹(F_{1}, F_{2}), see [65, p. 202].

Definition 2.6. A representation S is Cuntz–Pimsner covariant if ϑ(φ_{E}(a)) = φ_{F_{2}} (a) for all a ∈ A with φ_{E}(a) ∈ 𝕂(E).

By Proposition 2.3, a nondegenerate representation comes from an isomorphism of correspondences. We have already seen ϑ(φ_{E}(a)) = φ_{F_{2}} (a) for all a ∈ A in that case. So nondegenerate representations are Cuntz–Pimsner covariant. Conversely, let S be Cuntz–Pimsner covariant and assume that E is proper, that is, φ_{E}(A) ⊆ 𝕂(E). Let ⟨X⟩ denote the closed linear span of X. We have

⟨S(E)F_{1}⟩ ⊇ ⟨S(E)S(E)^{∗}F_{2}⟩ = ⟨ϑ(𝕂(E))F_{2}⟩

⊇ ⟨ϑ(φ_{E}(A))F_{2}⟩ = ⟨φ_{F_{2}} (A)F_{2}⟩ = ⟨F_{2}⟩

because φ_{F_{2}} is nondegenerate. Thus S is nondegenerate. □

For a proper correspondence E, we may now reformulate the universal property that defines its Cuntz–Pimsner algebra O_{E}: it is the universal C^{∗}-algebra for nondegenerate representations of E. Equivalently, O_{E} is the universal target for transformations from (A, E) to C^{∗}-algebras. The Cuntz–Pimsner algebra comes with a nondegenerate ^{∗}-homomorphism φ_{0} : A → O_{E} and a representation S_{0} : E → O_{E}, which is Cuntz–Pimsner covariant and thus nondegenerate. This is equivalent to a transformation from (A, E) to O_{E}; the underlying correspondence is O_{E} itself as a Hilbert O_{E}-module, with A acting via φ_{0}. The isomorphism u_{0} : E ⊗_{A} O_{E} ≅ O_{E} is the unitary that corresponds to S_{0} by Proposition 2.3. The transformation (O_{E}, u_{0}) has the following universal property: if (F, u) is another transformation from (A, E) to a C^{∗}-algebra B, then there is a unique representation ψ: O_{E} → 𝔹(F) for which u = u_{0}⊗_{ψ} id_{F}. Conversely, a representation ψ: O_{E} → 𝔹(F) provides a unitary u = u_{0}⊗_{ψ} id_{F} from E ⊗_{A}F ≅ E ⊗_{A}O_{E}⊗_{O_{E}} F to F ≅ O_{E} ⊗_{O_{E}} F. The pair (F, ψ) is the same as a correspondence from O_{E} to B. Thus transformations from (A, E) to B are the same as correspondences from O_{E} to B. What happens for a representation S: E → 𝔹(F) that does not satisfy the Cuntz–Pimsner covariance condition? The construction in the proof of Proposition 2.3 still gives a map u: E ⊗_{A} F → F, which is an A, B-bimodule map and isometric for the B-valued inner product. But this isometry u need not be unitary, not even adjointable. Thus allowing all Toeplitz representations replaces the unitary in the definition of a transformation by a possibly non-adjointable isometry.

Example 2.7. What goes wrong if E is not proper? Let us consider the simplest case, A = ℂ and E = ℓ^{2}(ℕ). In this case, no non-zero element of A acts by a compact operator, so there is no difference between the Cuntz–Pimsner and the Cuntz–Toeplitz algebra. A correspondence from A to B is the same as a Hilbert B-module. The Cuntz–Pimsner algebra O_{E} is the famous Cuntz algebra O_{∞} . The identity map on O_{∞} corresponds to a Cuntz–Pimsner covariant representation S_{0} : ℓ^{2}(ℕ) → O_{∞} , which maps the basis vector δ_{i} to the generating isometry S_{i}. The induced ^{∗}-homomorphism 𝕂(ℓ^{2}ℕ) → O_{∞} is degenerate, however, because O_{∞} is unital. It corresponds to the isometry of Hilbert O_{∞} -modules ℓ^{2}(ℕ) ⊗ O_{∞} ↪ O_{∞} , E_{ij} ⊗ x ↦ S_{i}xS_{j}^{∗}. If this were adjointable, its range would be of the form pO_{∞} for a projection p ∈ O_{∞} because O_{∞} is unital. Then [1]+p = p in K_{0}(O_{∞} ) because O_{∞} ⊕(ℓ^{2}(ℕ)⊗O_{∞} ) ≅ ℓ^{2}(ℕ)⊗O_{∞} , giving [1] = 0 in K_{0}(O_{∞} ), which is false.

Katsura’s definition of a relative Cuntz–Pimsner algebra only requires the Cuntz–Pimsner covariance condition on a certain ideal K ◁ A that acts on E by compact operators (see [41] or [42, Definition 3.4]). We may reformulate this as a partial nondegeneracy condition:

Proposition 2.8. Let A and B be C^{∗}-algebras, let E and F be correspondences from A to A and from A to B, respectively. Let K be an ideal in A that acts on E by compact operators. A representation S: E → 𝔹(F) satisfies the Cuntz–Pimsner covariance condition on K if and only if K · S(E)F = K · F. Equivalently, the isometry E ⊗_{A} F → F induced by S restricts to an isomorphism of correspondences KE ⊗_{A} F → KF.

Proof. Proposition 2.3 says that an isomorphism KE ⊗_{A} F → KF is equivalent to a nondegenerate representation KE → 𝔹(F, KF). Now apply Proposition 2.5 to the correspondences KE: K → A, F: A → B, and KF: K → B, so substitute K, A, B, KE, F, KF for A, B_{1}, B_{2}, E, F_{1}, F_{2}. Since we assume K to act by compact operators on E, the correspondence KE: K → A is always proper. So the nondegeneracy condition KE · F = KF is equivalent to the Cuntz–Pimsner covariance condition for the restriction of the left action of K to KF. That is, ϑ(φ_{E}(k))ξ = φ_{F} (k)ξ for all k ∈ K and ξ ∈ KF, with ϑ: 𝕂(E) → 𝔹(F) as in the proof of Proposition 2.5. It remains to show that this equality for all ξ ∈ KF implies the same equality for all ξ ∈ F: the latter is the usual coisometry condition for the ideal K. Let T_{k} := ϑ(φ_{E}(k)) − φ_{F} (k) for k ∈ K. Both T_{k} and T_{k}^{∗} = T_{k^{∗}} map F to KF = KE · F, and they vanish on KF by the above computation. Therefore, ⟨T_{k}ξ, T_{k}ξ⟩ = ⟨ξ, T_{k}^{∗}T_{k}ξ⟩ = 0 for all ξ ∈ F. □

3. Cuntz–Pimsner algebras of product systems over Ore monoids

Product systems over a monoid P were introduced by Fowler [33], inspired by previous definitions by Arveson [6] and Dinh [21]. The following data is equivalent to a product system in Fowler’s sense with the mild extra condition that each fibre be an essential left module over the unit fibre:

• a C^{∗}-algebra A;

• correspondences E_{p} from A to itself for all p ∈ P \ {1};

• isomorphisms of correspondences μ_{p,q}: E_{p} ⊗_{A} E_{q} → E_{pq} for all p, q ∈ P \{1}, which are associative, that is, the following diagram commutes for all p, q, t ∈ P:

\begin{matrix}E_{p} ⊗_{A} E_{q} ⊗_{A} E_{t} & E_{p} ⊗_{A} E_{qt} \\ E_{pq} ⊗_{A} E_{t} & E_{pqt} \end{matrix} id_{E_{p}} ⊗_{A} μ_{q,t} μ_{p,q} ⊗_{A} id_{E_{t}} μ_{p,qt} μ_{pq,t}

here we let E_{1} = A, and we let μ_{1,q} and μ_{p,1} be the isomorphisms A ⊗_{A} E_{q} ≅ E_{q} and E_{p} ⊗_{A} A ≅ E_{p} from the left and right A-module structures, respectively; this is needed to write down μ_{p,q} if p · q = 1 and to formulate the associativity condition for E_{p} ⊗_{A} E_{q} ⊗_{A} E_{t} → E_{pqt} if p · q = 1 or q · t = 1. Our main theorems will only hold if all correspondences E_{p} are proper. Then we speak of a proper product system over P.

Remark 3.1. A nondegenerate ^{∗}-homomorphism f: A → B gives a proper correspondence E_{f} from A to B: take E_{f} = B with A acting through f. For two composable nondegenerate ^{∗}-homomorphisms, we have a natural isomorphism E_{f} ⊗_{B} E_{g} ≅ E_{gf} . Due to this change in the order of products, an action of the opposite monoid P^{op} by ordinary nondegenerate ^{∗}-homomorphisms gives a product system over P. The Cuntz–Pimsner algebra of this product system is the same as the crossed product for the original action by endomorphisms, see [33, Section 3]. The change from P to P^{op} also explains why left Ore conditions are needed in [16, 50] to study actions of P by endomorphisms, while we will need right Ore conditions to study product systems over P.

Definition 3.2. Let (A, E_{p}, μ_{p,q}) be a product system over P. A transformation from it to a C^{∗}-algebra B consists of a correspondence F from A to B and isomorphisms of correspondences V_{p}: E_{p} ⊗_{A} F → F for p ∈ P \ {1}, such that for all p, q ∈ P \ {1}, the following diagram of isomorphisms commutes:

(3.3)  \begin{matrix}E_{p} ⊗_{A} E_{q} ⊗_{A} F & E_{p} ⊗_{A} F \\ E_{pq} ⊗_{A} F & F \end{matrix} ; V_{p} ; V_{pq} ; id_{E_{p}} ⊗_{A} V_{q} ; μ_{p,q} ⊗_{A} id_{F}

We let V_{1} be the canonical isomorphism A ⊗_{A} F ≅ F and use this in (3.3) if p · q = 1.

By Proposition 2.3, each isomorphism V_{p} corresponds to a nondegenerate representation S_{p}: E_{p} → 𝔹(F) of the correspondence E_{p}. By convention, S_{1} : E_{1} = A → 𝔹(F) is the representation of A that is part of the correspondence F. Equation (3.3) means that both maps around the square agree on all monomials ξ_{p} ⊗ ξ_{q} ⊗ η ∈ E_{p} ⊗_{A} E_{q} ⊗_{A} F. This amounts to the condition

S_{p}(ξ_{p}) · S_{q}(ξ_{q}) = S_{pq}(μ_{p,q}(ξ_{p} ⊗ ξ_{q})) for all ξ_{p} ∈ E_{p}, ξ_{q} ∈ E_{q},

which is standard for representations of product systems. Example 2.7 shows that we cannot expect enough transformations to exist unless our product system is proper. We assume this from now on. By Proposition 2.5, the nondegeneracy of the representations S_{p} is equivalent to the Cuntz–Pimsner covariance condition for all of them. Hence the universal property that defines the Cuntz–Pimsner algebra gives a natural bijection between correspondences from it to a C^{∗}-algebra B and transformations from the product system to B; this bijection leaves the underlying Hilbert module F unchanged. A transformation (F, V_{p}) gives unital, strictly continuous ^{∗}-homomorphisms

ϑ_{p}: 𝔹(E_{p}) → 𝔹(E_{p} ⊗_{A} F) ⟶^{≅} 𝔹(F), T ↦ V_{p}(T ⊗_{A} id_{F})V_{p}^{∗},

for all p ∈ P. Similarly, the isomorphisms μ_{p,q}: E_{p} ⊗_{A} E_{q} → E_{pq} induce nondegenerate ^{∗}-homomorphisms

φ_{p,q}: 𝕂(E_{p}) → 𝕂(E_{pq}), T ↦ μ_{p,q}(T ⊗_{A} id_{E_{q}})μ_{p,q}^{∗}. (3.4)

Since E_{q} is proper, φ_{p,q}(𝕂(E_{p})) is contained in 𝕂(E_{pq}). The commuting diagram (3.3) gives ϑ_{pq} ∘ φ_{p,q} = ϑ_{p} for all p, q ∈ P. This situation invites us to take a colimit (or inductive limit) of the C^{∗}-algebras 𝕂(E_{p}) along the maps φ_{p,q}. More precisely, let C_{P} be the category with object set P and arrow set P × P, where (p, q) is an arrow from p to pq, and where (pq, t) · (p, q) := (p, qt) for all p, q, t ∈ P.

Lemma 3.5. The maps p ↦ 𝕂(E_{p}) and (p, q) ↦ φ_{p,q} form a functor from C_{P} to the category of C^{∗}-algebras and nondegenerate ^{∗}-homomorphisms.

Proof. Functoriality means that φ_{pq,t} ∘ φ_{p,q} = φ_{p,qt} for all p, q, t ∈ P. This is equivalent to the associativity of μ. □

The diagram in Lemma 3.5 has a colimit in the category of C^{∗}-algebras and nondegenerate ^{∗}-homomorphisms. This colimit will act nondegenerately on F by its universal property. Therefore, it is part of the Cuntz–Pimsner algebra of the product system. In general, the colimit involves amalgamated free products, which make it rather intractable. To get a well-behaved Cuntz–Pimsner algebra, we assume that C_{P} is a filtered category in the following sense:

Definition 3.6 ([57, Section IX.1]). A category C is filtered if it is nonempty and

(F1) for any two objects x, y ∈ C_{0}, there are an object z ∈ C_{0} and arrows

g ∈ C(x, z) and h ∈ C(y, z); (F2) for any two parallel arrows g, h ∈ C(x, y), there are z ∈ C_{0} and k ∈

C(y, z) with kg = kh.

These conditions for C_{P} are equivalent to the following Ore conditions for P: (O1) for all x_{1}, x_{2} ∈ P, there are y_{1}, y_{2} ∈ P with x_{1}y_{1} = x_{2}y_{2}; (O2) if xy_{1} = xy_{2} for y_{1}, y_{2}, x ∈ P, then there is z ∈ P with y_{1}z = y_{2}z.

Definition 3.7. We call P a right Ore monoid if it has these two properties or, equivalently, C_{P} is filtered. We call P a left Ore monoid if P^{op} is a right Ore monoid.

Condition (O2) follows if P has cancellation. Both hold if P ⊆ G for a group G with P P^{−1} = G. Cancellative Ore monoids have already been considered by C^{∗}-algebraists; see, for instance, [16, 50]. We do not expect product systems over non-cancellative monoids to be very interesting (Lemma 4.16 hints strongly towards this), but it costs little extra effort to work in this greater generality. The monoid P is cancellative if and only if there is at most one arrow between any two objects in C_{P}. Then C_{P} is the category associated to a directed set, and colimits over C_{P} are the same as colimits over this directed set. Directed sets are familiar to analysts as the indexing sets for nets. Let P be a right Ore monoid. We may construct a group out of P by taking equivalence classes of formal quotients pq^{−1} := (p, q) for p, q ∈ P, where (p_{1}, q_{1}) ∼ (p_{2}, q_{2}) if there are t_{1}, t_{2} ∈ P with (p_{1}t_{1}, q_{1}t_{1}) = (p_{2}t_{2}, q_{2}t_{2}) (see also [13]). Condition (O1) implies that this relation is transitive and that products p_{1}q_{1}^{−1} · p_{2}q_{2}^{−1} may be rewritten as pq^{−1} by finding a common multiple of q_{1} and p_{2}: if q_{1}t_{1} = p_{2}t_{2}, then

p_{1}q_{1}^{−1} · p_{2}q_{2}^{−1} = (p_{1}t_{1})(q_{1}t_{1})^{−1} · (p_{2}t_{2})(q_{2}t_{2})^{−1} = (p_{1}t_{1})(q_{2}t_{2})^{−1} .

Hence we define the multiplication by [p_{1}, q_{1}]·[p_{2}, q_{2}] := [p_{1}t_{1}, q_{2}t_{2}] for t_{1}, t_{2} ∈ P with q_{1}t_{1} = p_{2}t_{2}. The conditions (O1) and (O2) imply that this is a well-defined group structure on G := P/∼ .

Example 3.8. All commutative monoids are Ore: we may take y_{1} = x_{2} and y_{2} = x_{1} in (O1) and z = x in (O2). Groups are also clearly Ore monoids. We may combine both classes as follows. Assume that G ⊆ P is a group such that xg_{1} = xg_{2} for x ∈ P, g_{1}, g_{2} ∈ G implies g_{1} = g_{2}; assume further that pG = Gp for all p ∈ P and p_{1}p_{2}G = p_{2}p_{1}G for all p_{1}, p_{2} ∈ P; roughly speaking, G is a normal subgroup of P such that P/G is commutative. We claim that such a monoid P is Ore. In (O1), given x_{1}, x_{2} ∈ P, we first take y_{1} = x_{2} and y_{2}^{0} = x_{1} as in the commutative case; then x_{1}y_{1}G = x_{2}y_{2}^{0}G, so there is g ∈ G with x_{1}y_{1} = x_{2}y_{2}^{0}g, so y_{2} = y_{2}^{0}g will do. In (O2), assume xy_{1} = xy_{2} for some x, y_{1}, y_{2} ∈ P. Then y_{1}xG = xy_{1}G = xy_{2}G = y_{2}xG, so there is g ∈ G with y_{1}x = y_{2}xg. Then xy_{1}x = xy_{2}xg = xy_{1}xg, which implies g = 1 by one of our assumptions. Thus y_{1}x = y_{2}x, so z = x works in (O2).

Example 3.9. Let R be a commutative, unital ring. Let R^{×} ⊆ R be the subset of all elements that are not zero divisors; this is the largest submonoid of (R, ·) that does not contain 0. The “ax + b-monoid” of R is the monoid P = R^{×} ⋉ R, consisting of pairs (a, b) ∈ R^{×} ⋉ R with the multiplication (a_{1}, b_{1}) · (a_{2}, b_{2}) = (a_{1}a_{2}, a_{1}b_{2} + b_{1}). This monoid is cancellative because we have taken away the zero divisors. It acts on R by (a, b) · x := ax + b. It contains the additive group (R, +) as a normal subgroup, and the quotient P/R is the commutative monoid (R^{×} , ·). Hence it is a special case of Example 3.8. C^{∗}-algebras associated to semigroups of this form have recently been studied by several authors, following Cuntz [15] and Li [54].

Example 3.10. Assume that P is cancellative and has a “norm” homomorphism N: P → (ℕ^{×} , ·) such that G := kerN is a subgroup. This is a special case of the situation in Example 3.8, with P/G ⊆ (ℕ^{×} , ·). One example of this type is the monoid 𝕄_{n}(ℤ)^{×} , the ring of integer matrices with non-zero determinant, with the determinant as norm: the kernel of the norm is the group Gl_{n}(ℤ). Another example is the monoid 𝕄_{n}(ℤ)^{×} ⋉ ℤ^{n}, with the determinant of the matrix part as norm and the semidirect product group Gl_{n}(ℤ) ⋉ ℤ^{n} as the kernel of the norm. More generally, we may replace 𝕄_{n}(ℤ) by the ring of integers in a simple algebra over ℚ; this also includes integer quaternion algebras. Semigroups of this form appear in generalisations of the Bost–Connes dynamical system, see [8, 14].

Example 3.11. The matrices of the form

h(a, b, c) := ( \begin{matrix}1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix})

for a, b, c ∈ ℕ form a noncommutative, cancellative monoid H_{ℕ} under matrix multiplication:

h(a_{1}, b_{1}, c_{1}) · h(a_{2}, b_{2}, c_{2}) = h(a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2} + a_{1}b_{2}).

This is an Ore monoid. To check the Ore condition (O1), pick h(a_{1}, b_{1}, c_{1}) and h(a_{2}, b_{2}, c_{2}) in H_{ℕ}. Let

a := max(a_{1}, a_{2}), b := max(b_{1}, b_{2}), c := max(c_{1}+a_{1}(b−b_{1}), c_{2}+a_{2}(b−b_{2})).

For i = 1, 2, let a_{i}^{⊥} := a − a_{i}, b_{i}^{⊥} := b − b_{i}, and c_{i}^{⊥} := c − c_{i} − a_{i}b_{i}^{⊥} ; then h(a_{i}, b_{i}, c_{i}) · h(a_{i}^{⊥} , b_{i}^{⊥} , c_{i}^{⊥} ) = h(a, b, c) for i = 1, 2, so we have found the desired common multiple. A similar formula works for the opposite monoid, so H_{ℕ} is both left and right Ore.

An inductive limit in the usual sense is the same as a colimit over the category associated to the poset (ℕ, ≤), which is easily seen to be filtered. Colimits over general filtered categories behave very much like inductive limits. This is well-known to category theorists. For the operator algebraists, we now assume that P is a countable Ore monoid, so that C_{P} is a countable filtered category. Then we may replace a colimit over C_{P} by an inductive limit over (ℕ, ≤):

Lemma 3.12. Let C be a countable filtered category. Then there is a sequence of objects (x_{n})_{n∈ℕ} and maps f_{n} ∈ C(x_{n−1} , x_{n}) such that for any object y of C there is n ∈ ℕ and an arrow y → x_{n}. Furthermore, if y → x_{n} and y → x_{m} are two such arrows, they become equal by composing with f_{N−1} ∘ ⋯∘ f_{n} : x_{n} → x_{n+1} → ⋯→ x_{N} and f_{N−1} ∘⋯ ∘f_{m} : x_{m} → x_{m+1} → ⋯→ x_{N} for sufficiently large N.

Such a sequence of objects and maps is called cofinal or final. More precisely, the functor (ℕ, ≤) → C given by the objects x_{n} and the maps f_{n} is called final in [57].

Proof. It is shown in [5] that any filtered category receives a cofinal functor from a directed (partially ordered) set. A partially ordered set is viewed as a category by putting a unique arrow x → y if x ≤ y, and no arrow otherwise. A category is of this form if and only if for any two objects there is at most one arrow between them. To simplify the proof, we first use [5] to reduce to a countable, directed set. The category C_{P} comes from a directed set if and only if P has cancellation. Let (y_{n})_{n∈ℕ} be an enumeration of the objects of C. We construct x_{n} for n ∈ ℕ inductively so that it receives maps from y_{1}, …, y_{n}. We start with x_{0} = y_{0}. Assume x_{i} and f_{i} have been constructed for i < n. Since C is filtered, there is an object x_{n} that receives maps from y_{n} and x_{n−1} . Let f_{n} be the arrow x_{n−1} → x_{n}. Since already x_{n−1} receives maps from y_{i} for i < n, so does x_{n} by composing with f_{n}. Thus every object y has a map to some x_{n}. Our simplifying assumption makes the second part of the lemma trivial. □

We now describe the colimit of the inductive system on C_{P} given by the C^{∗}-algebras 𝕂(E_{p}) for p ∈ P and the maps φ_{p,q} for p, q ∈ P defined in (3.4). We first do this quickly in the countable case. Then Lemma 3.12 allows us to choose a cofinal functor (ℕ, ≤) to C_{P}, that is, we get a pair of sequences (p_{n})_{n∈ℕ} and (q_{n})_{n∈ℕ} in P with p_{n+1} = p_{n}q_{n} for all n ∈ ℕ that is “cofinal” in C_{P}. The C^{∗}-algebras 𝕂(E_{p_{n}} ) and the nondegenerate ^{∗}-homomorphisms φ_{p_{n} ,q_{n}} : 𝕂(E_{p_{n}} ) → 𝕂(E_{p_{n} q_{n}} ) = 𝕂(E_{p_{n+1}} ) form an inductive system in the usual sense. Let O_{1} be its inductive limit C^{∗}-algebra. Cofinality implies that this inductive limit is also a colimit of the whole diagram on C_{P}. Now we give the more complicated construction without using Lemma 3.12, which also works in the uncountable case. Let

O_{⊔} := ⨆_{p∈P} 𝕂(E_{p}).

Let O_{∼} be the set of equivalence classes for the equivalence relation on O_{⊔} generated by the relations (x, p) ∼ (φ_{p,q}(x), pq) for all p, q ∈ P, x ∈ 𝕂(E_{p}).

Lemma 3.13. There is a unique ^{∗}-algebra structure for which all maps 𝕂(E_{p}) → O_{∼} are ^{∗}-homomorphisms, and the maximal C^{∗}-seminorm on O_{∼} exists. Let O_{1} be the resulting C^{∗}-completion of O_{∼} . If (F, V_{p}) is a transformation from (A, E_{p}, μ_{p,q}) to a C^{∗}-algebra B, then the resulting maps ϑ_{p}: 𝕂(E_{p}) → 𝔹(F) factor through a unique nondegenerate ^{∗}-homomorphism Θ: O_{1} → 𝔹(F).

Proof. Let x ∈ 𝕂(E_{p}), y ∈ 𝕂(E_{q}). There are t_{1}, t_{2} ∈ P with pt_{1} = qt_{2}. Then (x, p) ∼ (φ_{p,t_{1}} (x), pt_{1}) and (y, q) ∼ (φ_{q,t_{2}} (y), qt_{2}) both belong to the C^{∗}-algebra 𝕂(E_{pt_{1}} ) = 𝕂(E_{qt_{2}} ); this dictates what their sum or product should be in O_{∼} . If we choose t′_{1}, t′_{2} ∈ P with pt′_{1} = qt′_{2} instead, then we may find m_{1}, m_{2} ∈ P with pt′_{1}m_{1} = pt_{1}m_{2} and hence qt′_{2}m_{1} = qt_{2}m_{2}. If not yet t′_{2}m_{1} = t_{2}m_{2}, then we find n ∈ P with t′_{2}m_{1}n = t_{2}m_{2}n and replace m_{1}, m_{2} by m_{1}n, m_{2}n. Similarly, we achieve t′_{1}m_{1} = t_{1}m_{2}. Then multiplication with m_{2} and m_{1} will map our two choices of the sum or product to the same sum or product in 𝕂(E_{pt_{1} m_{2}} ), respectively. Thus the multiplication and addition on O_{∼} are well-defined. A similar argument shows that any finite subset of O_{∼} belongs to the image of 𝕂(E_{p}) in O_{∼} for some p ∈ P. Since the algebraic operations are defined using those in 𝕂(E_{p}), O_{∼} is a ^{∗}-algebra. By construction, this is the only ^{∗}-algebra structure for which all maps 𝕂(E_{p}) → O_{∼} are ^{∗}-homomorphisms.

The kernel of the map 𝕂(E_{p}) → O_{∼} is the union of the kernels of the ^{∗}-ho-momorphisms φ_{q,p}: 𝕂(E_{p}) → 𝕂(E_{pq}). Thus the image of 𝕂(E_{p}) in O_{∼} is the quotient by a union of closed ^{∗}-ideals. We equip it with the quotient seminorm, which is a C^{∗}-seminorm (there may be a nullspace because the union of ideals need not be closed). All these C^{∗}-seminorms on subalgebras of O_{∼} together are compatible with each other and thus define a C^{∗}-seminorm on O_{∼} . Since any ^{∗}-homomorphism between C^{∗}-algebras is contractive, this is the maximal C^{∗}-seminorm on the ^{∗}-algebra O_{∼} . Let O_{1} be the (Hausdorff) completion of O_{∼} for this C^{∗}-seminorm. This is a C^{∗}-algebra with ^{∗}-homomorphisms ϑ_{p}^{0}: 𝕂(E_{p}) → O_{1} for all p ∈ P that satisfy ϑ_{pq}^{0} ∘ φ_{p,q} = ϑ_{p}^{0} for all p, q ∈ P. Now take a transformation to B as above. The resulting maps ϑ_{p}: 𝕂(E_{p}) → 𝔹(F) satisfy ϑ_{pq} ∘ φ_{p,q} = ϑ_{p}. Hence the map ⨆ _{p∈G} ϑ_{p}: O_{⊔} → 𝔹(F) descends to a map f: O_{∼} → 𝔹(F). Since all ϑ_{p} are ^{∗}-homomorphisms, so is f. Since we took the maximal C^{∗}-seminorm on O_{∼} , we may extend f uniquely to a ∗-homomorphism Θ: O_{1} → 𝔹(F) with Θ ∘ ϑ_{p}^{0} = ϑ_{p} for all p ∈ P. □

Any functor (p_{n}, q_{n}): (ℕ, ≤) → C_{P} induces a ^{∗}-homomorphism from the inductive limit C^{∗}-algebra of the inductive system (𝕂(E_{p_{n}} ), φ_{p_{n} ,q_{n}} ) described above Lemma 3.13 to O_{1}. If the functor is cofinal, then this map is an isomorphism. Hence the simplified construction for countable P gives the same C^{∗}-algebra O_{1}. So far, we have described only a part of the Cuntz–Pimsner algebra of the product system. For a single endomorphism, this is the fixed-point subalgebra of the gauge action. We now describe the whole Cuntz–Pimsner algebra through a Fell bundle over the group completion G of P. Elements of G are equivalence classes of formal fractions p_{1}p_{2}^{−1} for p_{1}, p_{2} ∈ P, with p_{1}p_{2}^{−1} ∼ (p_{1}q)(p_{2}q)^{−1} . The fibre of the desired Fell bundle over G at 1 ∈ G is the C^{∗}-algebra O_{1} described above.

Definition 3.14. Fix g ∈ G. Let

R_{g} = {(p_{1}, p_{2}) ∈ P × P | p_{1}p_{2}^{−1} = g in G}

be its set of representatives. Let C_{P}^{g} be the category with object set R_{g} and arrow set R_{g} × P, where (p_{1}, p_{2}, q) is an arrow (p_{1}, p_{2}) → (p_{1}q, p_{2}q); the multiplication is (p_{1}q, p_{2}q, t) · (p_{1}, p_{2}, q) = (p_{1}, p_{2}, tq).

Lemma 3.15. The categories C_{P}^{g} for g ∈ G are filtered if P is an Ore monoid. The functor C_{P} → C_{P}^{1} that maps an object p in C_{P} to (p, p) and an arrow (p, q) in C_{P} to (p, p, q) is cofinal.

Proof. First we check that C_{P}^{g} is filtered. Let p = (p_{1}, p_{2}) and q = (q_{1}, q_{2}) be elements of R_{g}. We must prove two things. First, there should be arrows h: p → t and k: q → t with the same target t ∈ R_{g}. Secondly, if h, k: p ⇉ q are two parallel arrows, there is an arrow l: q → t for some object t such that l∘h = l∘k. Since p and q both represent g ∈ G, there are h, k ∈ P with p_{1}h = q_{1}k and p_{2}h = q_{2}k. Hence h: (p_{1}, p_{2}) → (p_{1}h, p_{2}h) and k: (q_{1}, q_{2}) → (q_{1}k, q_{2}k) have the same target, as desired. The second claim above is immediate from (O2): we may simply forget p_{2} and q_{2}.

The functor C_{P} → C_{P}^{1} is fully faithful. If (p_{1}, p_{2}) ∈ R_{1}, then there are q, h ∈ P with (p_{1}h, p_{2}h) = (q, q). Hence the functor C_{P} → C_{P}^{1} is cofinal. □

For (p_{1}, p_{2}) ∈ R_{g}, let O_{p_{1} ,p_{2}} := 𝕂(E_{p_{2}} , E_{p_{1}} ); for now, we view this as a Banach space. For q ∈ P, (p_{1}, p_{2}) ∈ R_{g}, we define a contraction

φ_{p_{1} ,p_{2} ,q} : 𝕂(E_{p_{2}} , E_{p_{1}} ) → 𝕂(E_{p_{2} q}, E_{p_{1} q}), T ↦ μ_{p_{1} ,q}(T ⊗_{A} id_{E_{q}})μ_{p_{2} ,q}^{∗}.

These maps form a functor from C_{P}^{g} to the category of Banach spaces with linear contractions. Since C_{P}^{g} is filtered by Lemma 3.15, the colimit O_{g} of this diagram may be constructed as in Lemma 3.13: first take the disjoint union of the Banach spaces O_{p_{1} ,p_{2}} for all (p_{1}, p_{2}) ∈ R_{g}; then divide out the relations given by the maps φ_{p_{1} ,p_{2} ,q}; this gives a vector space, and it inherits a canonical seminorm by taking the quotient seminorms on the images of 𝕂(E_{p_{2}} , E_{p_{1}} ); finally, take the completion to get O_{g}. If P is countable, then we may also use a cofinal sequence in C_{P}^{g} to describe the colimit as an inductive limit over (ℕ, ≤). Since the functor C_{P} → C_{P}^{1} is cofinal, the colimit of a diagram over C_{P}^{1} is the same as the colimit of its restriction to C_{P}. Hence the construction of O_{g} for g = 1 gives the same C^{∗}-algebra O_{1} as in Lemma 3.13, as suggested by our notation. If g_{1}, g_{2} ∈ G, (p_{1}, p_{2}) ∈ R_{g_{1}} , (p_{2}, p_{3}) ∈ R_{g_{2}} , then p_{1}p_{3}^{−1} = p_{1}p_{2}^{−1} · p_{2}p_{3}^{−1} = g_{1} · g_{2}, that is, (p_{1}, p_{3}) ∈ R_{g_{1} ·g_{2}}. The composition of compact operators gives a bounded bilinear map O_{p_{1} ,p_{2}} × O_{p_{2} ,p_{3}} → O_{p_{1} ,p_{3}} . These maps define a bounded bilinear map

O_{g_{1}} × O_{g_{2}} → O_{g_{1} g_{2}}

because for any (p′_{1}, p′_{2}), (p_{2}, p_{3}) in R_{g_{1}} ×R_{g_{2}} there are h, k ∈ P with p_{2}h = p′_{2}k, so that the composition is defined on O_{p′_{1} k,p′_{2} k} ×O_{p_{2} h,p_{3} h} , and these composition maps are compatible with the structure maps of the inductive systems. Similarly, taking adjoints gives maps O_{p_{1} ,p_{2}} → O_{p_{2} ,p_{1}} , T ↦ T^{∗}, for all p_{1}, p_{2} ∈ R_{g}; these maps induce an involution O_{g} → O_{g^{−1}}^{∗} . These multiplication maps and involutions on (O_{g})_{g∈G} give a Fell bundle over the group G. The resulting C^{∗}-algebra structure on its unit fibre O_{1} is the one already described in Lemma 3.13.

Theorem 3.16. Let P be an Ore monoid and let (A, E_{p}, μ_{p,q}) be a proper, nondegenerate product system over P. Its Cuntz–Pimsner algebra is isomorphic to the full sectional C^{∗}-algebra of the Fell bundle (O_{g})_{g∈G} described above.

Proof. Let C denote the Cuntz–Pimsner algebra of our product system. By construction, a nondegenerate^{∗}-homomorphism C → M(B) for a C^{∗}-algebra B is the same as a Cuntz–Pimsner covariant representation of our product system on B that is nondegenerate on the unit fibre A. The Cuntz–Pimsner covariance condition is equivalent to the nondegeneracy condition E_{p} · B = B for all p ∈ P by Proposition 2.5 because we assume all E_{p} to be proper and nondegenerate left A-modules, and the left A-action on B is nondegenerate as well. We are going to find a natural bijection between representations of the product system with E_{p} · B = B for all p ∈ P and representations of the Fell bundle (O_{g})_{g∈G} in M(B). By the universal property of the sectional C^{∗}-algebra of a Fell bundle, this gives a natural bijection between nondegenerate ^{∗}-homomorphisms C → M(B) and C^{∗}((O_{g})_{g∈G} ) → M(B), and this implies C ≅ C^{∗}((O_{g})_{g∈G} ). By Proposition 2.3, a representation of the product system that is nondegenerate in the above sense is equivalent to a transformation from (A, E_{p}, μ_{p,q}) to B with underlying Hilbert B-module B. We write F = B to be consistent with our previous notation. We already constructed ^{∗}-homomorphisms ϑ_{p}: 𝕂(E_{p}) → 𝔹(F) with ϑ_{pq} ∘ φ_{p,q} = ϑ_{p} for all p, q ∈ P. The same recipe gives linear contractions

ϑ_{p_{1} ,p_{2}} : 𝕂(E_{p_{2}} , E_{p_{1}} ) → 𝔹(F), T ↦ V_{p_{1}} (T ⊗ id_{F})V_{p_{2}}^{∗} .

These satisfy ϑ_{p_{1} q,p_{2} q} ∘ φ_{p_{1} ,p_{2} ,q} = ϑ_{p_{1} ,p_{2}} for all p_{1}, p_{2}, q ∈ P. Hence they induce maps Θ_{g}: O_{g} → 𝔹(F) on the Banach space inductive limits. Routine computations show that

(3.17) ϑ_{p_{2} ,p_{1}} (T)^{∗} = ϑ_{p_{1} ,p_{2}} (T^{∗}),  ϑ_{p_{1} ,p_{2}} (T) ∘ ϑ_{p_{2} ,p_{3}} (T_{2}) = ϑ_{p_{1} ,p_{3}} (T ∘ T_{2})

for all p_{1}, p_{2}, p_{3} ∈ P, T ∈ 𝕂(E_{p_{2}} , E_{p_{1}} ), T_{2} ∈ 𝕂(E_{p_{3}} , E_{p_{2}} ). Hence the maps Θ_{g} form a representation of the Fell bundle (O_{g})_{g∈G} . Conversely, a representation of the Fell bundle (O_{g})_{g∈G} gives maps

𝕂(E_{p_{2}} , E_{p_{1}} ) → 𝔹(F)

that satisfy (3.17). For p_{2} = 1, there is a canonical isomorphism 𝕂(E_{p_{2}} , E_{p_{1}} ) ≅ E_{p_{1}} because E_{1} = A. Hence the Fell bundle representation gives maps S_{p}: E_{p} → 𝔹(F). Since A = 𝕂(E_{1}) ⊆ O_{1}, the conditions of a Fell bundle representation imply that the maps S_{p} form a representation of the product system. Since the maps 𝕂(E_{p}) → O_{1} → 𝔹(F) are nondegenerate, S_{p}(E_{p})F ⊇ 𝕂(E_{p})F = F. This gives the desired bijection between Fell bundle representations and Cuntz–Pimsner covariant representations of the product system and finishes the proof. □

Theorem 3.16 is similar to [49, Theorem 3.8], only the assumptions on the product system differ slightly. Unlike in [49], we assume the product system to be nondegenerate, but allow the left action to be non-injective. Similar ideas are used in [16, Section 4] to dilate certain actions of Ore semigroups by endomorphisms to group actions on a larger algebra. In fact, actions by endomorphisms are a special case of product systems by Remark 3.1. To reduce the dilation results in [16] or [50] to Theorem 3.16, another step is missing: checking when the Fell bundle in Theorem 3.16 comes from an ordinary group action by automorphisms. We refrain from doing this because, from our point of view, saturated Fell bundles are already actions of the underlying group.

Proposition 3.18. The Fell bundle (O_{g})_{g∈G} is saturated if E_{p} is a full Hilbert A-module for each p ∈ P.

Proof. Let g ∈ G and let p ∈ P. We want to show that the image of 𝕂(E_{p}) in O_{1} is contained in the space of right inner products from O_{g}. There is (p_{1}, p_{2}) ∈ R_{g} and q ∈ P with pq = p_{1}. The image of 𝕂(E_{p}) in O_{1} is contained in the image of 𝕂(E_{pq}) = 𝕂(E_{p_{1}} ). Since E_{p_{1}} and E_{p_{2}} are full, both 𝕂(E_{p_{1}} ) and 𝕂(E_{p_{2}} ) are Morita–Rieffel equivalent to A and hence equivalent to each other. The equivalence between them is 𝕂(A, E_{p_{1}} ) ⊗_{A} 𝕂(E_{p_{2}} , A) ≅ 𝕂(E_{p_{2}} , E_{p_{1}} ). Hence the latter is a full Hilbert bimodule over 𝕂(E_{p_{2}} ) and 𝕂(E_{p_{1}} ). Since 𝕂(E_{p_{2}} , E_{p_{1}} ) ⊆ O_{g}, it follows that the right inner products from O_{g} give a dense subspace of 𝕂(E_{p_{1}} ) in O_{1}. Thus the Fell bundle (O_{g})_{g∈G} is saturated. □

Remark 3.19. The criterion in Proposition 3.18 is not necessary for rather trivial reasons. If the left A-actions on E_{p} are not faithful, then it may happen that O_{1} = 0. Since this has nothing to do with E_{p} being full as a right Hilbert module, the Fell bundle (O_{g}) may be saturated although not all E_{p} are full.

Saturated Fell bundles over a group G are interpreted as actions of G by correspondences in [12]. Long before, it was known that one may replace a saturated Fell bundle (O_{g})_{g∈G} with unit fibre O_{1} by an action of G by automorphisms on a C^{∗}-algebra Õ_{1} that is Morita–Rieffel equivalent to O_{1}: this is the Packer– Raeburn Stabilisation Trick. Non-saturated Fell bundles over G are interpreted in [11] as actions of G by Hilbert bimodules, that is, partial Morita–Rieffel equivalences. The analogue of the Packer–Raeburn Stabilisation Trick says that any Fell bundle, saturated or not, is equivalent to an action of G by partial ∗-isomorphisms. A saturated Fell bundle over G may, of course, be restricted to a product system over P. Which product systems are of this form?

Proposition 3.20. A proper product system over P is the restriction of a saturated Fell bundle over G if and only if each E_{p} is an A, A-imprimitivity bimodule, that is, each E_{p} is a full right Hilbert A-module and the left action is by an isomorphism A ≅ 𝕂(E_{p}). The saturated Fell bundle over G is unique up to isomorphism.

Proof. In a saturated Fell bundle over G, each E_{g} is an imprimitivity bimodule. Conversely, assume that E_{p} is an imprimitivity bimodule for each p ∈ P. Then all the maps 𝕂(E_{p}) → 𝕂(E_{pq}) in our inductive system are isomorphisms, so that the inductive limit O_{1} is isomorphic to A = 𝕂(E_{1}). Similarly, O_{p} ≅ E_{p} for all p ∈ P. Thus our product system is the restriction to P of a Fell bundle over G. Since all E_{p} are assumed to be full, this Fell bundle is saturated by Proposition 3.18. Now start with a saturated Fell bundle (O_{g})_{g∈G} , restrict it to P, and then go back to a Fell bundle over G. The maps 𝕂(E_{p_{2}} , E_{p_{1}} ) → 𝕂(E_{p_{2} q}, E_{p_{1} q}) are isomorphisms for all p_{1}, p_{2}, q ∈ P, so the inductive systems that give the fibres of the new Fell bundle are constant. Thus the colimit O_{g} is canonically isomorphic to 𝕂(E_{p_{2}} , E_{p_{1}} ) for any (p_{1}, p_{2}) ∈ R_{g}, and our construction of a Fell bundle from (E_{p})_{p∈P} reproduces the original Fell bundle up to isomorphism. Hence the product system on P determines the saturated Fell bundle over G uniquely up to isomorphism. □

A non-saturated Fell bundle over G need not give a proper product system on P: this requires E_{p} to be full as a left Hilbert A-module for each p ∈ P.

Theorem 3.21. If A is nuclear or exact, then so is O_{1}. If A is nuclear and the group G generated by P is amenable, then the Cuntz–Pimsner algebra O is nuclear. If A is exact and G is amenable, then O is exact.

Proof. The first claim follows because O_{1} is an inductive limit of C^{∗}-algebras Morita–Rieffel equivalent to A and because nuclearity and exactness are hereditary under Morita–Rieffel equivalence and filtered inductive limits. The other statements follow from Theorem 3.16 and general results about nuclearity and exactness of Fell bundle C^{∗}-algebras. First, if the group is amenable, then any Fell bundle over it has the approximation property, which implies that the full and reduced sectional C^{∗}-algebras coincide (see [24]). The exactness of the reduced sectional C^{∗}-algebra is proved in [25], assuming exact unit fibre and an exact group. The nuclearity of the full sectional C^{∗}-algebra is proved in [1], assuming nuclear unit fibre and an amenable group. □

Next we describe the K-theory of the unit fibre O_{1} of our Fell bundle. Since E_{p} is a proper correspondence from A to A, it gives an element [E_{p}] ∈ KK_{0}(A, A) with zero operator F. This gives a map

(E_{p})_{∗}: K_{∗}(A) → K_{∗}(A);

here K_{∗}(A) denotes the ℤ/2-graded K-theory of A comprising both K_{0} and K_{1}. The Kasparov product of [E_{p}] and [E_{q}] is [E_{p}⊗_{A} E_{q}] with zero operator; since the Fredholm operator is irrelevant, this case of the Kasparov product is easy. The isomorphisms μ_{p,q} now show that [E_{p}] ⊗_{A} [E_{q}] = [E_{pq}] and hence (E_{q})_{∗} ∘ (E_{p})_{∗} = (E_{pq})_{∗} for all p, q ∈ P. The order of p and q is changed here because ⊗_{A} is the composition product in KK in reverse order. Hence our product system over P gives an action of P^{op} on K_{∗}(A). We view this as a right module structure over the monoid ring ℤ[P]. The group ring ℤ[G] is a left module over ℤ[P].

Theorem 3.22. Let P be an Ore monoid and let (A, E_{p}, μ_{p,q}) give a proper, nondegenerate product system over P. Assume also that all E_{p} are full right Hilbert A-modules. Then the K-theory of O_{1} is isomorphic to K_{∗}(A)⊗_{ℤ[P]}ℤ[G] as a right ℤ[G]-module; here we use the canonical right module structure on K_{∗}(A) ⊗_{ℤ[P]} ℤ[G] by right multiplication and the module structure on K_{∗}(O_{1}) induced by the saturated Fell bundle (O_{g})_{g∈G} .

Proof. It is well-known that K-theory is compatible with inductive limits. This extends to colimits over countable filtered categories by Lemma 3.12. We leave it to the reader interested in uncountable monoids to check that the result remains true for arbitrary filtered colimits. Hence K_{∗}(O_{1}) is the colimit of the diagram over C_{P} that maps p ∈ P to K_{∗}(𝕂(E_{p})) and (p, q): p → pq to (φ_{p,q})_{∗}: K_{∗}(𝕂(E_{p})) → K_{∗}(𝕂(E_{pq})). Since E_{p} is a full Hilbert bimodule, it gives a Morita–Rieffel equivalence from 𝕂(E_{p}) to A. This correspondence with zero operator F is a cycle for KK_{0}(𝕂(E_{p}), A). This is a KK-equivalence: the inverse is the inverse imprimitivity bimodule 𝕂(E_{p}, A) with zero operator F. We use this KK-equivalence to identify K_{∗}(𝕂(E_{p})) ≅ K_{∗}(A) for all p ∈ P. Composing the maps

K_{∗}(A) ⟶_{≅} K_{∗}(𝕂(E_{p})) −−−−→^{(φ_{p,q} )_{∗}} K_{∗}(𝕂(E_{pq})) ⟶_{≅} K_{∗}(A)

requires composing three KK_{0}-cycles with zero operator F, which amounts to tensoring the underlying correspondences. Identifying E_{pq} ≅ E_{p} ⊗_{A} E_{q} as in the definition of φ_{p,q}, we see that this composite is [E_{q}]. Thus the inductive system with colimit K_{∗}(O_{1}) is isomorphic to the inductive system with entries K_{∗}(A) at all p ∈ P, where the arrow (p, q): p → pq in C_{P} induces the map (E_{q})_{∗}: K_{∗}(A) → K_{∗}(A). Define a diagram of left ℤ[P]-modules over C_{P} by taking the free module ℤ[P] at all objects and letting q: p → pq act by δ_{x} ↦ δ_{xq} for all x, p, q ∈ P. The colimit of this diagram of modules is isomorphic to ℤ[G] by mapping ℤ[P] ∋ δ_{x} at the object p of C_{P} to δ_{xp^{−1}} ∈ ℤ[G]. Hence M ⊗_{ℤ[P]}ℤ[G] for a right ℤ[P]-module M is the colimit of the diagram over C_{P} with entries M ⊗_{ℤ[P]} ℤ[P] ≅ M and with q: p → pq acting by m ↦ m · q for all m ∈ M, p, q ∈ P. Now compare this with our description of the inductive system that computes K_{∗}(O_{1}) to get K_{∗}(O_{1}) ≅ K_{∗}(A) ⊗_{ℤ[P]} ℤ[G]. Since the Hilbert modules E_{p} are full, the Fell bundle (O_{g})_{g∈G} is saturated by Proposition 3.18. Then each O_{g} is a proper correspondence from O_{1} to itself and hence gives a class [O_{g}] in KK_{0}(O_{1}, O_{1}). This induces maps (O_{g})_{∗}: K_{∗}(O_{1}) → K_{∗}(O_{1}). Since we have a saturated Fell bundle, we have O_{g} ⊗_{O_{1}} O_{h} ≅ O_{gh}. Therefore, g ↦ (O_{g})_{∗} defines a representation of G^{op} on the Abelian group K_{∗}(O_{1}); we view this as a right ℤ[G]-module structure. To describe this action, it suffices to compute, for p ∈ P, how (O_{g})_{∗} acts on the image of K_{∗}(𝕂(E_{p})) in K_{∗}(O_{1}) under the map K_{∗}(ϑ_{p}^{0}) induced by ϑ_{p}^{0}: 𝕂(E_{p}) → O_{1}. First choose (p_{1}, p_{2}) ∈ R_{g} and then q ∈ P with pq = p_{1}. Then K_{∗}(ϑ_{p}^{0}) = K_{∗}(ϑ_{p_{1}}^{0} ) ∘ K_{∗}(φ_{p,q}), so it suffices to describe how (O_{g})_{∗} acts on the image of K_{∗}(𝕂(E_{p_{1}} )). The map 𝕂(E_{p_{2}} , E_{p_{1}} ) → O_{g} shows that (ϑ_{p_{1}}^{0} )^{∗}(O_{g}) ≅ 𝕂(E_{p_{2}} , E_{p_{1}} ) ⊗_{ϑ_{p_{2}}^{0}} O_{1} as correspondences from 𝕂(E_{p_{1}} ) to O_{1}. Now compose these correspondences with the KK-equivalences between 𝕂(E_{p_{1}} ) and A. Then we see that (O_{g})_{∗} acts on the entry K_{∗}(A) at p_{1} in the inductive system describing K_{∗}(O_{1}) by sending it to the same entry at p_{2}. Right multiplication by g = p_{1}p_{2}^{−1} in K_{∗}(A) ⊗_{ℤ[P]} ℤ[G] has the same effect. Thus the action of G on K_{∗}(O_{1}) induced by the Fell bundle corresponds to the one by right multiplication on K_{∗}(A) ⊗_{ℤ[P]} ℤ[G]. □

By the Packer–Raeburn Stabilisation Trick, there is a G-action by automorphisms on the stabilisation Õ_{1} := O_{1} ⊗ 𝕂(L^{2}G) such that the (full) crossed product G ⋉ Õ_{1} is Morita–Rieffel equivalent to the Cuntz-Pimsner algebra O (this follows from [12, Corollary 5.5]). Thus computing the K-theory of the Cuntz–Pimsner algebra becomes a matter for the (full) Baum–Connes conjecture for G with certain coefficients.

For a-T-menable groups, the Baum–Connes assembly map is known to be an isomorphism for all coefficients, also for the full crossed product (see [36]). The meaning of the Baum–Connes conjecture here is that we may compute K_{∗}(O) by topological means from K_{∗}(O|_{H}), the section algebras for restrictions of (O_{g})_{g∈G} to all finite subgroups H. These topological means may be expressed as a spectral sequence, and it can be quite hard to perform this computation in practice. At least, the results above show that the computation for a Cuntz– Pimsner algebra over P is not more difficult than in the special case of an action of G by automorphisms. For instance, let P = (ℕ^{k}, +) for k ∈ ℕ. Then G = ℤ^{k}, and the computation of K_{∗}(O) is a matter of iterating the Pismner–Voiculescu sequence k times. We will consider a concrete case where a K-theory computation along these lines is feasible in Section 3.3. Already two iterations may be very hard because the boundary maps for the second iteration are not determined by the original data.

Remark 3.23. The iteration of the Pimsner–Voiculescu sequence that we get is equivalent to one by Deaconu in [20]. This is because the Pimsner–Voiculescu sequence for ℤ-actions can be obtained from the Cuntz–Toeplitz algebra of the product system over ℕ associated to the ℤ-action.

Our theory contains the case of semigroup crossed products for actions of left Ore monoids by endomorphisms by Remark 3.1. In that case, (E_{p})_{∗}: K_{∗}(A) → K_{∗}(A) is simply the map induced by the underlying endomorphism. The K-theory computation for semigroup C^{∗}-algebras in [16] also uses the Baum–Connes isomorphism for the group G. In the situation of [16], a dilation of the semigroup action to an action of G on a larger C^{∗}-algebra is easy to write down by hand, giving a direct route to K_{∗}(O_{1}). But then extra assumptions on the action of G on K_{∗}(O_{1}) are needed to compute K_{∗}(O).

3.1. Making left actions faithful. Let (A, E_{p}, μ_{p,q}) be a proper product system over an Ore monoid P. Taking suitable quotients of A and E_{p}, we are going to construct another product system (A′, E′_{p}, μ′_{p,q}) with the same nondegenerate representations and hence the same Cuntz–Pimsner algebra, such that the left actions φ′_{p}: A′ → 𝔹(E′_{p}) are injective for all p ∈ P. For p ∈ P, let φ_{p}: A → 𝕂(E_{p}) denote the left action map and let I_{p} := kerφ_{p}. Recall the maps φ_{p,q}: 𝕂(E_{p}) → 𝕂(E_{pq}) for p, q ∈ P. Since φ_{p,q} ∘ φ_{p} = φ_{pq}, we have I_{p} ⊆ I_{pq} for all p, q ∈ P. Since C_{P} is filtered, this implies that the ideals I_{p} form a directed set of ideals in A. Thus I := \overline{⋃ _{p∈P} I_{p}} is another ideal in A. We let A′:= A/I and E′_{p} := E_{p} ⊗_{A} A′ ≅ E_{p} / (E_{p} · I).

Lemma 3.24. The induced left action A → 𝕂(E′_{p}) factors through A′, and the isomorphism μ_{p,q}: E_{p} ⊗_{A} E_{q} ⟶^{∼} E_{pq} descends to an isomorphism of correspondences μ′_{p,q}: E′_{p} ⊗_{A} E′_{q} ⟶^{∼} E′_{pq}. This gives a product system (A′, E′_{p}, μ′_{p,q}).

Proof. Let p ∈ P. To prove that the induced left A-module structure on E′_{p} descends to A′, we must show that IE_{p} ⊆ E_{p}I. Since P is an Ore monoid, the subset pP is cofinal in P, so ⋃ _{q∈P} I_{pq} is still dense in I. Thus it suffices to prove I_{pq}E_{p} ⊆ E_{p}I for all p, q ∈ P. We will prove the following more precise result:

(3.25)  I_{pq} = {a ∈ A | aE_{p} ⊆ E_{p}I_{q}}.

Let ξ ∈ E_{p}. We have ξ ⊗_{A} η = 0 in E_{p} ⊗_{A} E_{q} for all η ∈ E_{q} if and only if

0 = ⟨ξ ⊗ η_{1}, ξ ⊗ η_{2}⟩ = ⟨η_{1}, φ_{q}(⟨ξ, ξ⟩_{A})η_{2}⟩

for all η_{1}, η_{2} ∈ E_{q}, if and only φ_{q}(⟨ξ, ξ⟩_{A}) = 0, if and only if ⟨ξ, ξ⟩_{A} ∈ I_{q}. We claim that this is equivalent to ξ ∈ E_{p} · I_{q}. Since I_{q} is an ideal, we have ⟨ξ, ξ⟩_{A} ∈ I_{q} for ξ ∈ E_{p} · I_{q}. Conversely, if ⟨ξ, ξ⟩_{A} ∈ I_{q}, then the closure of ξ · A in E_{p} is a Hilbert I_{q}-module containing ξ, and thus it is nondegenerate as a right I_{q}-module, so that ξ ∈ E_{p}I_{q}. Hence ξ ⊗_{A} η = 0 in E_{p} ⊗_{A} E_{q} for all η ∈ E_{q} if and only ξ ∈ E_{p}I_{q}. Now let a ∈ A. Then aξ ∈ E_{p}I_{q} for all ξ ∈ E_{p} if and only if aξ ⊗_{A} η = 0 for all ξ ∈ E_{p}, η ∈ E_{q}, if and only if the left action by a vanishes on E_{p} ⊗_{A} E_{q} ≅ E_{pq}. This is equivalent to a ∈ I_{pq}. This finishes the proof of (3.25). In turn, this implies that the left A-module structure on E′_{p} descends to A′. Now

E′_{p} ⊗_{A′} E′_{q} = E_{p} ⊗_{A} A′⊗_{A′} E_{q} ⊗_{A} A′ = E_{p} ⊗_{A} E_{q} ⊗_{A} A′ ≅ E_{pq} ⊗_{A} A′ = E′_{pq}.

This gives the multiplication maps μ′_{p,q}. From another point of view, μ′_{p,q} is the map on the quotient spaces E′_{p} and E′_{q} induced by μ_{p,q}. Hence these maps inherit associativity from the maps μ_{p,q}, so we have constructed a product system. □

Theorem 3.26. The product system (A′, E′_{p}, μ′_{p,q}) has faithful left action maps A′ → 𝕂(E′_{p}), and it has the same nondegenerate representations as the original system. Hence it also has the same Cuntz–Pimsner algebra.

Proof. Fix p ∈ P. An operator on E_{p} induces the zero operator on E_{p}/E_{p}I_{q} ≅ E_{p} ⊗_{A} (A/I_{q}) if and only if it maps E_{p} into E_{p}I_{q}. Thus (3.25) shows that the map φ_{p}: A → 𝕂(E_{p}) descends to an injective ^{∗}-homomorphism A/I_{pq} ↪ 𝕂(E_{p}/E_{p}I_{q}). The C^{∗}-algebras A/I_{pq} and 𝕂(E_{p}/E_{p}I_{q}) for q ∈ P form inductive systems indexed by the filtered category C_{P}, and the maps A/I_{pq} ↪ 𝕂(E_{p}/E_{p}I_{q}) form a morphism of inductive systems, consisting of injective maps. It follows that the induced map between the inductive limits \underrightarrow{lim}A/I_{pq} = A/\overline{⋃ _{q∈P} I_{pq}} = A′ and \underrightarrow{lim}𝕂(E_{p}/E_{p}I_{q}) = 𝕂(E′_{p}) is injective as well. That is, the left action A′ → 𝕂(E′_{p}) is faithful. Now let ϑ′: A′ → 𝔹(F) and S′_{p}: E′_{p} → 𝔹(F) for p ∈ P give a nondegenerate representation of the product system (A′, E′_{p}, μ′_{p,q}). Composing with the quotient maps A → A′ and E_{p} → E′_{p} then gives a nondegenerate representation of (A, E_{p}, μ_{p,q}). We claim that any nondegenerate representation (ϑ, S_{p}) of (A, E_{p}, μ_{p,q}) factors through the quotient maps A → A′ and E_{p} → E′_{p} and thus comes from a unique representation (ϑ′, S′_{p}). This gives a bijection on the level of nondegenerate representations and thus an isomorphism of Cuntz– Pimsner algebras because they are universal for nondegenerate representations by Proposition 2.5. Recall the maps ϑ_{p}: 𝕂(E_{p}) → 𝔹(F) with ϑ_{pq} ∘ φ_{p,q} = ϑ_{p} for all p, q ∈ P. In particular, ϑ = ϑ_{p} ∘ φ_{p}: A → 𝔹(F), so ϑ must vanish on I_{p}. Since this holds for all p ∈ P, we get ϑ|_{I} = 0, so ϑ factors through the quotient map A → A′. Since S_{p}(ξ)^{∗}S_{p}(ξ) = ϑ(⟨ξ, ξ⟩_{A}) for ξ ∈ E_{p} and ⟨ξ, ξ⟩ ∈ I for ξ ∈ E_{p} · I, we also get S_{p}(ξ) = 0 for ξ ∈ E_{p} · I. Thus S_{p} factors through E′_{p}. □

Proposition 3.27. If the maps A → 𝕂(E_{p}) are injective for all p ∈ P, then so are the induced maps φ_{p,q,t} : 𝕂(E_{q}, E_{p}) → 𝕂(E_{qt}, E_{pt}) for p, q, t ∈ P and the maps 𝕂(E_{q}, E_{p}) → O to the Cuntz–Pimsner algebra.

Proof. We assume that I_{p} = {0} for all p ∈ P. The proof of (3.25) shows that ξ ∈ E_{p} satisfies ξ ⊗_{A} η = 0 in E_{p} ⊗_{A} E_{q} for all η ∈ E_{q} if and only if ξ = 0. Hence the maps φ_{p,q,t} are injective. Since O_{g} ⊆ O is the filtered colimit of the spaces 𝕂(E_{q}, E_{p}), this implies the same for the maps 𝕂(E_{q}, E_{p}) → O_{g} ⊆ O. □

3.2. What happens without the Ore conditions? We now consider an example of a monoid without the Ore conditions where we can, nevertheless, describe the Cuntz–Pimsner algebra by hand. Let F_{n}^{+} be the free monoid on n generators, n ≥ 2. Elements in F_{n}^{+} are finite words in the letters a_{1}, …, a_{n}, including the empty word. This monoid violates the Ore conditions: there are no words w_{1}, w_{2} ∈ F_{n}^{+} with a_{1}w_{1} = a_{2}w_{2}. A proper product system over F_{n}^{+} is equivalent to a C^{∗}-algebra A with proper correspondences E_{i} from A to itself for i = 1, …, n, without any further data or conditions: given this data, we may define E_{w} for a word w by composing the correspondences for the letters in w, and we use the canonical multiplication maps between them.

Proposition 3.28. Let A be a C^{∗}-algebra and let E_{i} for 1 ≤ i ≤ n be proper correspondences from A to A. Let O_{i} be the Cuntz–Pimsner algebra of E_{i} for 1 ≤ i ≤ n. The Cuntz–Pimsner algebra of the resulting product system over F_{n}^{+} is the amalgamated free product of the Cuntz–Pimsner algebras O_{i} over A.

Proof. Let D be another C^{∗}-algebra and let G be a Hilbert module over D. A nondegenerate representation of our product system over F_{n}^{+} on G is already determined by what it does on the correspondences E_{i}, and E_{i} may act by arbitrary nondegenerate representations because F_{n}^{+} is a free monoid. A nondegenerate representation of E_{i} is equivalent to a representation of the Cuntz– Pimsner algebra O_{i} by Proposition 2.5. Since all these representations give the same representation when we compose with the canonical map A → O_{i}, we get a representation of the amalgamated free product of the C^{∗}-algebras O_{i} over A. Conversely, a representation of this free product gives nondegenerate representations of the correspondences E_{i} and thus of A, and it gives the same representation on A for each i. This data may be extended to a nondegenerate representation of the product system over F_{n}^{+} . □

Free products with amalgamation are, unfortunately, rather large and complicated. In particular, they are almost never nuclear or exact. Thus we view Proposition 3.28 as a negative result: it tells us that we should not expect Cuntz–Pimsner algebras for proper product systems over F_{n}^{+} to have a nice structure. Standard assumptions in the theory of Cuntz–Toeplitz and Cuntz– Pimsner algebras are that the underlying semigroup be “quasi-lattice-ordered” and the product system “compactly aligned,” see [33]. Both assumptions are satisfied for proper product systems over F_{n}^{+} . If two elements in F_{n}^{+} have an upper bound, they have a least upper bound because two elements in F_{n}^{+} only have an upper bound if one of them is a subword of the other, and then the longer of the two is a least upper bound. Hence Cuntz–Pimsner algebras of compactly aligned product systems over quasi-lattice-ordered monoids need not be tractable.

3.3. Higher-rank Doplicher–Roberts algebras. In this section, we consider higher-rank analogues of the C^{∗}-algebras introduced by Doplicher and Roberts in [22]. The Doplicher–Roberts C^{∗}-algebras were an important motivation for Kumjian, Pask, Raeburn and Renault when they defined graph C^{∗}-algebras in [48]. Our higher-rank analogue is constructed from a compact Lie group G and finite-dimensional representations π_{1}, …, π_{k} of G; in addition, we need a representation ρ: G → U(H) on a Hilbert space H that contains each irreducible representation of G. Different choices for ρ will, however, give Morita–Rieffel equivalent C^{∗}-algebras, so we consider ρ to be auxiliary data only. From the above data, we are going to construct a product system over the commutative monoid (ℕ^{k}, +) and then take its Cuntz–Pimsner algebra. The case k = 1 is considered in [48]. For m = (m_{1}, …, m_{k}) ∈ ℕ^{k}, we form the representation

π^{m} := π_{1}^{⊗m_{1}} ⊗ ⋯⊗ π_{k}^{⊗m_{k}} : G → U(V^{m});

here V^{m} denotes the finite-dimensional Hilbert space on which π^{m} acts. There are canonical unitary operators μ_{m_{1} ,m_{2}} : V^{m_{1}} ⊗V^{m_{2}} ≅ V^{m_{1} +m_{2}} that intertwine the representations π^{m_{1}} ⊗ π^{m_{2}} and π^{m_{1} +m_{2}} , and which satisfy the properties of a symmetric monoidal category. Let E_{m} ⊆ 𝕂(H, V^{m} ⊗ H) be the space of all compact intertwining operators between the representations ρ and π^{m} ⊗ ρ. Define multiplication maps E_{m_{1}} × E_{m_{2}} → E_{m_{1} +m_{2}} by mapping (T_{1}, T_{2}) to the composite intertwining operator

ρ −→^{T_{2}} π^{m_{2}} ⊗ ρ −−−→^{1⊗T_{1}} π^{m_{2}} ⊗ π^{m_{1}} ⊗ ρ −−−−−⟶^{μ_{m_{2} ,m_{1}} ⊗1} π^{m_{1} +m_{2}} ⊗ ρ;

this composite is compact because T_{2} is compact.

Lemma 3.29. The multiplication above is associative.

Proof. Let m_{1}, m_{2}, m_{3} ∈ ℕ^{k} and let T_{i} ∈ E_{m_{i}} for i = 1, 2, 3. Then the products (T_{1}T_{2})T_{3} and T_{1}(T_{2}T_{3}) are equal to the composite operators

ρ  π^{m_{3}} ⊗ ρ  π^{m_{3}} ⊗ π^{m_{2}} ⊗ ρ  π^{m_{3}} ⊗ π^{m_{2}} ⊗ π^{m_{1}} ⊗ ρ

π^{m_{3} +m_{2}} ⊗ ρ  π^{m_{3} +m_{2}} ⊗ π^{m_{1}} ⊗ ρ

T_{3} π^{m_{3} +m_{2} +m_{1}} ⊗ ρ 1⊗T_{2} 1⊗1⊗T_{1} μ_{m3 ,m2} ⊗1 μ_{m3 ,m2} ⊗1 1⊗1⊗T_{1} μ_{m3 +m2 ,m1} ⊗1

because μ_{m_{3} +m_{2} ,m_{1}} ∘ (μ_{m_{3} ,m_{2}} ⊗ 1) = μ_{m_{3} ,m_{2} +m_{1}} ∘ (1 ⊗ μ_{m_{2} ,m_{1}} ). □

The unit fibre E_{0} is the C^{∗}-algebra of all compact intertwining operators of ρ. The multiplication maps above turn each E_{m} into an E_{0}-bimodule. We define an E_{0}-valued right inner product on E_{m} by ⟨T_{1}, T_{2}⟩ := T_{1}^{∗}T_{2} for T_{1}, T_{2} ∈ E_{m}. This turns each E_{m} into a correspondence from the C^{∗}-algebra E_{0} to itself.

Lemma 3.30. The C^{∗}-algebra 𝕂(E_{m}) is isomorphic to the C^{∗}-algebra of compact intertwiners of the representation π^{m} ⊗ ρ, acting on E_{m} by left multiplication. More generally, 𝕂(E_{m_{1}} , E_{m_{2}} ) is isomorphic to the space of compact intertwiners π^{m_{1}} ⊗ ρ → π^{m_{2}} ⊗ ρ.

Proof. The map sending |T_{1}⟩⟨T_{2}|∈ 𝕂(E_{m}) for T_{1}, T_{2} ∈ E_{m} to the intertwiner T_{1}T_{2}^{∗}: π^{m} ⊗ ρ → π^{m} ⊗ ρ extends to a ^{∗}-homomorphism from 𝕂(E_{m}) to the C^{∗}-algebra of compact intertwiners of π^{m} ⊗ ρ. Since |T_{1}⟩⟨T_{2}|T_{3} = T_{1}T_{2}^{∗}T_{3}, this representation is faithful. It remains to show that it is surjective. Any compact intertwiner on π^{m} ⊗ ρ may be approximated by linear combinations of intertwiners with irreducible range because the representation π^{m} ⊗ ρ, like any representation of G, is a direct sum of irreducible representations. Since any irreducible representation of G occurs in ρ, any intertwiner with irreducible range factors through the representation ρ. Thus we may write it as T_{1}T_{2}^{∗} for T_{1}, T_{2} ∈ E_{m}. This shows that 𝕂(E_{m}) is mapped onto the C^{∗}-algebra of compact intertwiners of π^{m} ⊗ ρ. The same argument still works in the more general case of 𝕂(E_{m_{1}} , E_{m_{2}} ). □

If T ∈ E_{0}, then the induced operator 1 ⊗ T: π^{m} ⊗ ρ → π^{m} ⊗ ρ is compact as well because the representation π^{m} has finite dimension. Thus the correspondence E_{m} is proper by Lemma 3.30.

Lemma 3.31. The multiplication maps induce unitary operators E_{m_{1}} ⊗_{E_{0}} E_{m_{2}} → E_{m_{1} +m_{2}} .

Proof. It is routine to check that the map E_{m_{1}} × E_{m_{2}} → E_{m_{1} +m_{2}} defined above preserves the inner products, so it gives an isometry E_{m_{1}} ⊗_{E_{0}} E_{m_{2}} → E_{m_{1} +m_{2}} . This induces a ^{∗}-homomorphism 𝕂(E_{m_{1}} ) → 𝕂(E_{m_{1} +m_{2}} ), T ↦ T ⊗ 1. In terms of Lemma 3.30, this is given by the map T ↦ T ⊗ 1 from compact intertwiners of π^{m_{1}} ⊗ ρ to compact intertwiners of π^{m_{1}} ⊗ π^{m_{2}} ⊗ ρ ≅ π^{m_{1} +m_{2}} ⊗ ρ. This ∗-homomorphism on compact operators is nondegenerate. Hence the underlying isometry E_{m_{1}} ⊗_{E_{0}} E_{m_{2}} → E_{m_{1} +m_{2}} must be surjective. □

Lemma 3.31 says that the correspondences E_{m} with the above multiplication maps form an essential product system over the commutative monoid (ℕ^{k}, +). As remarked above, Lemma 3.30 implies that this product system is proper. Since any irreducible representation occurs in ρ, the Hilbert E_{0}-module E_{m} is full and carries a faithful left E_{0}-action.

Definition 3.32. The Cuntz–Pimsner algebra of the product system (E_{m})_{m∈ℕ^{k}} over (ℕ^{k}, +) is the higher-rank Doplicher–Roberts algebra for the representations π_{1}, …, π_{n} of G, relative to ρ.

Lemma 3.33. The higher-rank Doplicher–Roberts algebras for different choices of ρ are canonically Morita–Rieffel equivalent.

Proof. Let ρ and ρ′ be two representations of G that contain all irreducible representations. Lemma 3.30 identifies E_{0}^{ρ} and E_{0}^{ρ′} with the C^{∗}-algebras of compact intertwiners on the representations ρ and ρ′, respectively. Let F_{ρρ′} be the space of all compact intertwining operators ρ′ → ρ. This is a full Hilbert bimodule for E_{0}^{ρ} and E_{0}^{ρ′}. Furthermore, we may naturally identify both F_{ρρ′} ⊗_{E_{0}^{ρ′}} E_{m}^{ρ′} and E_{m}^{ρ} ⊗_{E_{0}^{ρ}} F_{ρρ′} with the space of compact intertwiners from ρ′ to π^{m} ⊗ρ. These identifications provide a Morita equivalence between the product systems for ρ and ρ′and thus induce a Morita–Rieffel equivalence between their Cuntz–Pimsner algebras. □

To clarify the link to previous constructions, take k = 1 and let ρ be the direct sum of all irreducible representations with multiplicity 1. Then the C^{∗}-algebra E_{0} of compact intertwiners of ρ is C_{0}(Ĝ). Since k = 1, our product system is determined by the single self-correspondence of C_{0}(Ĝ) given by E_{1}. Such a self-correspondence is equivalent to a graph with vertex set Ĝ. Since the left action on E_{1} is faithful and E_{1} is proper and full, our graph has neither sources nor sinks and no infinite emitters. Hence our absolute Cuntz–Pimsner algebra agrees with the relative one used to define graph C^{∗}-algebras. Our Doplicher–Roberts algebra is exactly the graph C^{∗}-algebra considered in [48, Section 7]. As shown there, the C^{∗}-algebra defined by Doplicher and Roberts in [22] is isomorphic to a full corner in this graph C^{∗}-algebra (assuming that each irreducible representation of G occurs in π^{m} for some m ∈ ℕ).

Remark 3.34. For k > 1, it seems unlikely that our higher-rank Doplicher– Roberts algebras are higher-rank graph C^{∗}-algebras. For k = 1, any product system over (ℕ^{k}, +) with unit fibre of the form C_{0}(V) for a discrete set V (“vertices”) gives a higher-rank graph C^{∗}-algebra. For k > 1, this fails: we also need the multiplication isomorphisms in the product system to be given by permutation matrices in some chosen bases for our self-correspondences.

Let D denote our higher-rank Doplicher–Roberts algebra. The general theory above applies here and shows that D = ⨁ _{m∈ℤ^{k}} D_{m} is the section C^{∗}-algebra of a Fell bundle (D_{m})_{m∈ℤ^{k}} over ℤ^{k}, where D_{m} is the filtered colimit of the Banach spaces 𝕂(E_{a}, E_{a+m} ) for a ∈ ℕ^{k} with a + m ∈ ℕ^{k}. Here Lemma 3.30 identifies this Banach space with the space of compact intertwiners π^{a}⊗ρ → π^{a+m} ⊗ρ. In particular, the zero fibre D_{0} is the inductive limit of the system of C^{∗}-algebras 𝕂(E_{a}) for a ∈ ℕ^{k}; the sequence given by a = (a_{1}, a_{1}, …, a_{1}) for a_{1} ∈ ℕ is cofinal in ℕ^{k}, so we may as well take this sequence. For each a ∈ ℕ^{k}, 𝕂(E_{a}) is a C^{∗}-algebra of compact operators, hence a direct sum of matrix algebras. The summands are in bijection with Ĝ because all irreducible representations of G occur in ρ and hence in π^{a} ⊗ ρ. In particular, 𝕂(E_{a}) is Morita–Rieffel equivalent to C^{∗}(G) for each a ∈ ℕ^{k}, and it is an AF-algebra whose K-theory is equal to the K-theory K_{0}(C^{∗}(G)) of the group C^{∗}-algebra of G and hence to the representation ring of G. Concretely, its elements are functions Ĝ → (ℤ, +), and the positive cone in K_{0}(𝕂(E_{a})) consists of all functions Ĝ → (ℕ, +). A countable inductive limit of AF-algebras remains an AF-algebra, so D_{0} is AF. We compute its K-theory. The group K_{0}(C^{∗}(G)) is a commutative ring through the tensor product of representations: the representation ring of G. The map

K_{0}(C^{∗}(G)) ≅ K_{0}(𝕂(E_{a})) → K_{0}(𝕂(E_{a+m} )) ≅ K_{0}(C^{∗}(G))

induced by the canonical ^{∗}-homomorphism 𝕂(E_{a}) → 𝕂(E_{a+m} ) is the multiplication with [π^{m}] in the ring structure on K_{0}(C^{∗}(G)). Hence

K_{0}(D_{0}) = \underrightarrow{lim}(K_{0}(C^{∗}(G)) −−−→^{[π^{1_{k}} ]} K_{0}(C^{∗}(G)) −−−→^{[π^{1_{k}} ]} K_{0}(C^{∗}(G)) → ⋯),

where [π^{1_{k}} ] denotes multiplication with the class of the representation π_{1} ⊗ ⋯⊗ π_{k} in the representation ring. This inductive limit is the localisation of the representation ring K_{0}(C^{∗}(G)) of G in which we invert [π^{1_{k}} ]. Our K-theory computation determines D_{0} uniquely up to Morita–Rieffel equivalence by Elliott’s classification of AF-algebras. Each D_{m} is a Hilbert bimodule from D_{0} to itself, which acts on K_{0} by multiplication with the class of π^{m}. From this information, it is sometimes possible to compute the K-theory of D. We do not pursue this in general but merely consider one special case where we can completely describe the higher-rank Doplicher–Roberts algebra.

Theorem 3.35. Let G = SU(n) for n ≥ 2 and let π_{i} = Λ^{i}(ℂ^{n}) ∈ Ĝ for i = 1, …, n − 1 be the exterior powers of the standard representation on ℂ^{n}. The associated rank-n−1 Doplicher–Roberts algebra D is purely infinite, simple, separable, nuclear and in the bootstrap class, and has K_{0}(D) = ℤ, K_{1}(D) = 0.

Kirchberg’s Classification Theorem implies that D is isomorphic to the Cuntz algebra O_{∞} , but we would not expect this isomorphism to be constructible.

Proof. The representations π_{1}, …, π_{n−1} ∈ Ĝ are the fundamental representations of SU(n), that is, they are irreducible and generate a ring isomorphism ℤ[x_{1}, …, x_{n−1} ] ↦ R(G), x_{i} ↦ [π_{i}].

Our Fell bundle description of D shows that it is stably isomorphic to a crossed product of an action of ℤ^{n−1} on an AF-algebra. Hence it is separable, nuclear and in the bootstrap class. We compute the K-theory of D by iterating the Pimsner–Voiculescu exact sequence n−1 times. We write D⋊ℤ^{i} for the crossed product with ℤ^{i} ⊆ ℤ^{n−1} , although this is really a Fell bundle section algebra; the action by automorphisms only occurs on a stably isomorphic C^{∗}-algebra. The AF-algebra D_{0} has K_{0}(D_{0}) = ℤ[x_{1}^{±1} , …, x_{n−1}^{±1} ] because the representation ring of G is isomorphic to the polynomial algebra ℤ[x_{1}, …, x_{n−1} ] with x_{i} = [π_{i}] and localising at the elements x_{1}, …, x_{n−1} simply adjoins their inverses. The elements 1 − x_{1}, …, 1 − x_{n−1} form a regular sequence in this algebra; that is, for each i, multiplication by 1 − x_{i+1} is injective on the quotient ℤ[x_{1}^{±1} , …, x_{n−1}^{±1} ]/(1 − [x_{1}], …, 1 − [x_{i}]) by the ideal generated by 1 − [x_{1}], …, 1 − [x_{i}]. This is what allows us to compute the K-theory by repeated application of the Pimsner–Voiculescu exact sequence. In each step, we are supposed to consider the kernel and cokernel of the map 1 − α_{i} on K_{∗}(D_{0} ⋊ ℤ^{i−1} ), where α_{i} is induced by the action of the ith factor of ℤ on D_{0} ⋊ ℤ^{i−1} . By induction, we show that K_{1}(D_{0} ⋊ ℤ^{i}) vanishes and that K_{0}(D_{0} ⋊ ℤ^{i}) is the quotient ring ℤ[x_{1}^{±1} , …, x_{n−1}^{±1} ]/(1 − x_{1}, …, 1 − x_{i}) ≅ ℤ[x_{i+1}^{±1} , …, x_{n−1}^{±1} ]. This is clear for i = 0. In each induction step, we use the Pimsner–Voiculescu exact sequence. Since we have a regular sequence, multiplication by 1 − x_{i} is injective on ℤ[x_{1}^{±1} , …, x_{n−1}^{±1} ]/(1 − x_{1}, …, 1 − x_{i−1} ). Thus K_{1}(D_{0}⋊ℤ^{i}) vanishes and K_{0}(D_{0}⋊ℤ^{i}) is ℤ[x_{1}^{±1} , …, x_{n−1}^{±1} ]/(1−x_{1}, …, 1− x_{i}). After n − 1 steps, we get K_{1}(D) = 0 and

K_{0}(D) ≅ ℤ[x_{1}^{±1} , …, x_{n−1}^{±1} ]/(1 − x_{1}, …, 1 − x_{n−1} ) ≅ ℤ.

Thus D has the asserted K-theory. Next we prove that D is simple. We do not claim that the AF-algebra D_{0} is simple. The crossed product D_{0} ⋊ ℤ^{1}, however, is simple by [48, Corollary 7.3] because the representation π_{1} of G on ℂ^{n} is faithful. This crossed product is just a rank-1 Doplicher–Roberts algebra, hence stably isomorphic to a graph algebra. The graph algebra description of D_{0} ⋊ ℤ^{1} shows also that it is purely infinite. The K-theory computation above shows that D_{0} ⋊ ℤ^{1} has K-theory isomorphic to ℤ[x_{2}^{±1} , …, x_{n−1}^{±1} ], where the automorphism associated to π_{2}^{m_{2}} ⋯ π_{n−1}^{m_{n−1}} acts by multiplication with x_{2}^{m_{2}} ⋯ x_{n−1}^{m_{n−1}} . Since this is never the identity map, none of these automorphisms can be inner. Since D_{0} ⋊ ℤ^{1} is simple, separable and purely infinite, the (reduced) crossed product by the group ℤ^{n−2} remains simple and purely infinite by [46, Lemma 10]. Since the stabilisation (D_{0} ⋊ℤ^{1})⋊ℤ^{n−2} of D is simple and purely infinite, so is D itself. □

There is another way to construct higher-rank Doplicher–Roberts algebras using the comultiplication Δ: C^{∗}(G) → C^{∗}(G) ⊗ C^{∗}(G), which is defined by λ_{g} ↦ λ_{g} ⊗ λ_{g} for the standard multipliers λ_{g} of C^{∗}(G) for g ∈ G. This comultiplication turns C^{∗}(G) into a discrete quantum group. Our comultiplication is only a morphism, that is, its image is only in the multiplier algebra of C^{∗}(G). We know, however, that (C^{∗}(G) ⊗ 1) · Δ(C^{∗}(G)) = C^{∗}(G) ⊗ C^{∗}(G). Let π: C^{∗}(G) → 𝕄_{n}(ℂ) be some finite-dimensional representation of C^{∗}(G) or, equivalently, of G. We get a morphism

C^{∗}(G) ⟶^{Δ} C^{∗}(G) ⊗ C^{∗}(G) −−−→^{π⊗id} 𝕄_{n} ⊗ C^{∗}(G) ≅ 𝕂(C^{∗}(G)^{n}).

Since (C^{∗}(G) ⊗ 1) · Δ(C^{∗}(G)) = C^{∗}(G) ⊗ C^{∗}(G), this morphism has values in 𝕂(C^{∗}(G)^{n}). Thus any finite-dimensional representation π of C^{∗}(G) induces a proper correspondence E(π) from C^{∗}(G) to itself. Let π_{i}: C^{∗}(G) → 𝕄_{n_{i}} (ℂ) for i = 1, 2 be two finite-dimensional representations and let π_{1} ⊗ π_{2} : C^{∗}(G) → 𝕄_{n_{1} n_{2}} (ℂ) be their tensor product representation. The coassociativity of Δ gives an isomorphism of correspondences

E(π_{1}) ⊗_{C^{∗} (G)} E(π_{2}) ≅ E(π_{1} ⊗ π_{2}).

Hence the obvious intertwining unitary π_{1} ⊗ π_{2} ≅ π_{2} ⊗ π_{1} gives canonical isomorphisms E(π_{1}) ⊗_{C^{∗} (G)} E(π_{2}) ≅ E(π_{2}) ⊗_{C^{∗} (G)} E(π_{1}). The category of representations of G with the tensor product of representations and the obvious associators and commuters π_{1} ⊗π_{2} ≅ π_{2} ⊗π_{1} is a symmetric monoidal category. Therefore, k representations π_{1}, …, π_{k} of G give a product system over the monoid (ℕ^{k}, +) with fibres

E(m_{1}, …, m_{k}) := E(π_{1}^{⊗m_{1}} ⊗ ⋯⊗ π_{k}^{⊗m_{k}} )

and with the canonical isomorphisms

E(m_{1}, …, m_{k}) ⊗_{C^{∗} (G)} E(m′_{1}, …, m′_{k}) ≅ E(m_{1} + m′_{1}, …, m_{k} + m′_{k}).

We claim that this product system is the same as the one constructed above if ρ is the regular representation. A first point is that the C^{∗}-algebra of compact intertwiners of ρ is canonically isomorphic to C^{∗}(G), acting by the right regular representation. Hence E_{0}^{ρ} ≅ C^{∗}(G). Furthermore, ρ absorbs every other representation by the Fell absorption principle: π ⊗ ρ is canonically isomorphic to a sum of n copies of ρ if π has dimension n; the intertwiners L^{2}(G, ℂ^{n}) ↔ L^{2}(G, ℂ^{n}) are given by pointwise multiplication with the matrix π_{g} at g ∈ G. Hence we may identify C^{∗}(G)^{n} canonically with the Hilbert E_{0}^{ρ}-module of compact intertwiners ρ → π ⊗ ρ. These identifications provide an isomorphism between our product systems because the tensor product of representations of G is induced by the comultiplication Δ.

4. Actions of Ore monoids on spaces

Now let X be a locally compact, Hausdorff space and let A = C_{0}(X). Since any automorphism of A comes from a homeomorphism on X, we may turn an action of a group G on A by automorphisms into an action of G on the space X and form a transformation groupoid G ⋉ X. The crossed product G ⋉ C_{0}(X) is canonically isomorphic to the groupoid C^{∗}-algebra of G ⋉ X. When is there such a groupoid model for the Cuntz–Pimsner algebra of a self-correspondence on A?

As a counterexample, consider a Hermitian vector bundle over X. It gives a proper self-correspondence from A to itself by taking the Hilbert module of sections with its usual inner product and the left action by pointwise multiplication. The resulting Cuntz–Pimsner algebra is a locally trivial field of C^{∗}-algebras over X with Cuntz algebras as fibres. Such C^{∗}-algebras are classified by Dădărlat in [18] in terms of certain cohomology groups. Unless the field of C^{∗}-algebras over X is particularly simple, it seems to have no natural groupoid model. Therefore, we restrict attention to self-correspondences of C_{0}(X) that are induced by topological correspondences (see [43]). We define product systems of such topological correspondences in the obvious fashion, so that they induce a product system of C^{∗}-correspondences. We will build a “transformation groupoid” for a proper product system of topological correspondences and show that its groupoid C^{∗}-algebra is isomorphic to the Cuntz–Pimsner algebra of the product system. Our transformation groupoid construction is similar in spirit to the boundary path groupoid of Yeend [79] for a higher-rank topological graph, that is, for the case P = ℕ^{k} for some k ≥ 1. Yeend’s construction, however, depends on special features of ℕ^{k}. In contrast, our construction depends on the properness of the product systems. A topological correspondence between two spaces X and Y is given by a third space M with two maps r: M → X and s: M → Y. We want to turn this into a C^{∗}-correspondence from C_{0}(X) to C_{0}(Y). There are two ways to do this. First, we may assume that s is a local homeomorphism; this is Katsura’s definition of a topological correspondence in [43]. Other names for this are continuous graphs (see [19]) or polymorphisms (see [7, 17]). Secondly, we may add extra data, namely, a family of measures (λ_{x})_{x∈X} on the fibres of s; this is what Muhly and Tomforde call a topological quiver in [60]. The family of measures (λ_{x}) is equivalent to a transfer operator for s in the notation of Exel [26], or a Markov operator in the notation of [39]. A topological correspondence gives a topological quiver when combined with a suitably normalised family of counting measures on the (discrete) fibres of s. A topological quiver (M, r, s, λ_{x}) gives a C^{∗}-correspondence E_{r,M,s} over C_{0}(X): complete C_{c}(M) with respect to the C_{0}(X)-valued inner product

⟨ξ_{1}, ξ_{2}⟩(x) := ∫_{s^{−1} (x)} (\overline{ξ_{1}}ξ_{2})(y) dλ_{x}(y)

for ξ_{1}, ξ_{2} ∈ C_{c}(M), x ∈ X; the left and right module structures are (fξ)(m) := f(r(m))ξ(m) and (ξf)(m) := ξ(m)f(s(m)) for all m ∈ M, f ∈ C_{0}(X), ξ ∈ C_{c}(M). In particular, this construction applies to topological correspondences, where we always take the family of counting measures.

Proposition 4.1. The C^{∗}-correspondence E_{r,M,s} is proper if and only if r is proper and s is a local homeomorphism. In that case, the isomorphism class of E_{r,M,s} does not depend on (λ_{x}), so we may always use the family of counting measures. The C^{∗}-correspondence E_{r,M,s} is full if and only if s is surjective.

Proof. The C^{∗}-correspondence E_{r,M,s} is proper if and only if φ^{−1} (𝕂(E_{r,M,s} )) = C_{0}(X). In the notation of [60, Definition 3.14], all vertices are finite emitters. [60, Corollary 3.12] shows that this happens if and only if r is proper and s is a local homeomorphism. Let (λ_{x}) and (λ′_{x}) be two families of measures that make (M, r, s) into a topological quiver. Since both λ_{x} and λ′_{x} have the same discrete subset s^{−1} (x) as support, they are equivalent, say, λ′_{x} = f_{x} · λ_{x} for a unique function f_{x}: s^{−1} (x) → (0, ∞). The functions f_{x} may be pieced together to a function f: M → (0, ∞). The continuity of (λ_{x}) and (λ′_{x}) implies that f is a continuous function. Hence multiplication with \sqrt{f} is a unitary operator between the Hilbert modules over C_{0}(X) associated to the two families of measures. This unitary also intertwines the left actions, which are by multiplication operators. It is routine to check that E_{r,M,s} is full if and only if s is surjective. □

Definition 4.2. A topological correspondence is called proper if r is proper and s is a local homeomorphism.

Lemma 4.3. Consider two topological correspondences

X ⟵^{r_{1}} M_{1} ⟶^{s_{1}} X ⟵^{r_{2}} M_{2} ⟶^{s_{2}} X.

Define M := M_{1} ×_{s_{1} ,X,r_{2}} M_{2}, r: M → X, (m_{1}, m_{2}) ↦ r_{1}(m_{1}), s: M → X, (m_{1}, m_{2}) ↦ s_{2}(m_{2}). Then

E_{r_{1} ,M_{1} ,s_{1}} ⊗_{C_{0} (X)} E_{r_{2} ,M_{2} ,s_{2}} ≅ E_{r,M,s} .

If r_{1} and r_{2} are proper, so is r. If s_{1} and s_{2} are surjective, so is s.

Proof. The first part is routine to prove and holds even for topological quivers, see [60, Lemmas 6.1–4]. The statements about proper and surjective maps are easy as well; they amount to the statement that tensor products of proper or full C^{∗}-correspondences are again proper or full, respectively. □

Proposition 4.1 says that the C^{∗}-correspondence associated to a topological quiver is proper if and only if we are dealing with a proper topological correspondence; the family of measures does not matter. We restrict attention to proper topological correspondences from now on. The notion of a “topological graph algebra” interprets a topological correspondence as a “topological graph,” where vertices and (oriented) edges form topological spaces. This interpretation, however, fails to elucidate the lack of symmetry between r and s in the construction of the C^{∗}-correspondence. Another interpretation is that a topological correspondence (r, M, s) is a multi-valued map from Y to X, where r(m) ∈ X for m ∈ s^{−1} (y) are the possible values at y ∈ Y. If s is a local homeomorphism and r is proper, then the subset of values r(s^{−1} (y)) of y is discrete. The interpretation as a multivalued map breaks down, however, if there are different m, m′ ∈ M with s(m) = s(m′) and r(m) = r(m′). We suggest the following more dynamical interpretation of a (proper) topological correspondence. We consider points in M as possible developments or, briefly, stories. Each story m ∈ M assumes a certain initial situation s(m) ∈ Y and leads to a certain ending r(m) ∈ X. Several stories may have the same initial situation and ending. How does this interpretation account for the assumptions that s be a local homeomorphism and r be proper? That s is a local homeomorphism means the following: if we modify the initial situation s(m) of a story m a little bit, then there is a unique story m_{x} close to m with initial situation x. Roughly speaking, m_{x} describes how “the same” story would go in a slightly different initial situation, and fits our intuition of story-telling. That r is proper means that, given a compact set of possible endings, the set of stories with such an ending is also compact. This is a rather technical finiteness condition on the space of possible stories. It ensures that the space of complete histories defined below is locally compact.

Definition 4.4. Let P be a monoid. An action of P on X by proper topological correspondences consists of the following data:

• proper topological correspondences (M_{p}, r_{p}, s_{p}) from X to X for p ∈

P \ {1};
• homeomorphisms σ_{p,q}: M_{pq} → M_{p} ×_{s_{p} ,X,r_{q}} M_{q} for p, q ∈ P \ {1}.

Let M_{1} = X and r_{1} = s_{1} = id_{X}, and let σ_{p,1} and σ_{1,q} be the canonical homeomorphisms M_{p} ≅ M_{p} ×_{s_{p} ,X,id_{X}} X and M_{q} ≅ X ×_{id_{X} ,X,r_{q}} M_{q} for p, q ∈ P. For an action of P, we require the diagram

(4.5)  \begin{matrix}M_{p} ×_{X} M_{q} ×_{X} M_{t} & M_{pq} ×_{X} M_{t} \\ M_{p} ×_{X} M_{qt} & M_{pqt} \end{matrix} ; σ_{p,q} ×_{X} id_{M_{t}} ; id_{M_{p}} ×_{X} σ_{q,t} ; σ_{pq,t} ; σ_{p,qt}

to commute for all p, q, t ∈ P \ {1} (since pq = 1 or qt = 1 is possible, we have to define (M_{1}, s_{1}, r_{1}), σ_{1,q} and σ_{p,1} for this condition to make sense). This diagram commutes automatically if p = 1, q = 1 or t = 1, so our assumption implies that it commutes for all p, q, t ∈ P.

Example 4.6. An action of ℕ^{k} on a countable discrete set X by proper topological correspondences is equivalent to a row-finite rank-k graph by [34]. The Cuntz–Pimsner algebra that we shall attach to this data is not always the higher-rank graph C^{∗}-algebra, however, because we do not incorporate Katsura’s modification of the Cuntz–Pimsner algebra into our definition. See also [67].

We fix an action of P on X by proper topological correspondences as above. The proper topological correspondences (M_{p}, r_{p}, s_{p}) induce proper C^{∗}-correspondences E_{p} from C_{0}(X) to itself for p ∈ P \ {1}, and we let E_{1} := C_{0}(X). The homeomorphisms σ_{p,q} induce isomorphisms of C^{∗}-correspondences

μ_{p,q}: E_{p} ⊗_{C_{0} (X)} E_{q} → E_{pq}

for p, q ∈ P \ {1} by Lemma 4.3, and we let μ_{1,q} and μ_{p,1} be the canonical isomorphisms. The diagram (4.5) ensures the associativity of these multiplication maps μ_{p,q} for all p, q, t ∈ P \ {1} (even if pq = 1 or qt = 1); associativity is automatic if p = 1, q = 1 or t = 1. So an action of P on X by topological correspondences induces a proper product system over P with unit fibre C_{0}(X), as expected. The defining property of the fibre product means that σ_{p,q} = (r_{p,q}, s_{p,q}) for two continuous maps

r_{p,q}: M_{pq} → M_{p}, s_{p,q}: M_{pq} → M_{q}

with s_{p} ∘ r_{p,q} = r_{q} ∘ s_{p,q}. Since σ_{p,1} and σ_{q,1} are the canonical maps,

s_{p,1} = s_{p}, r_{p,1} = id_{M_{p}} , s_{1,q} = id_{M_{q}} , r_{1,q} = r_{q}

for all p, q ∈ P. The associativity condition (4.5) is equivalent to

(4.7)  r_{p,q} ∘ r_{pq,t} = r_{p,qt}, s_{p,q} ∘ r_{pq,t} = r_{q,t} ∘ s_{p,qt}, s_{q,t} ∘ s_{p,qt} = s_{pq,t}.

Lemma 4.8. The maps r_{p,q} are proper and the maps s_{p,q} are local homeomorphisms. If all s_{p} are surjective, then so are the maps s_{p,q}.

Proof. The map r_{p,q} is the composite of the homeomorphism σ_{p,q} and the coordinate projection M_{p} ×_{s_{p} ,X,r_{q}} M_{q} → M_{p}. This coordinate projection is proper if r_{q} is proper because properness is hereditary under this type of fibre products. Similarly, the map s_{p,q} is the composite of the homeomorphism σ_{p,q} and the coordinate projection M_{p} ×_{s_{p} ,X,r_{q}} M_{q} → M_{q}; the latter inherits the property of being surjective or a local homeomorphism from s_{p}. □

We interpret elements of P as a (multi-dimensional) kind of time, and elements of M_{p} as stories of length p ∈ P; a story m ∈ M_{p} starts in the situation s_{p}(m) and ends in r_{p}(m). The maps r_{p,q}: M_{pq} → M_{p} and s_{p,q}: M_{pq} → M_{q} cut a story m of length pq into two stories of length p and q: its ending m_{1} = r_{p,q}(m) ∈ M_{p} and its beginning m_{2} = s_{p,q}(m) ∈ M_{q}. These satisfy s_{p}(m_{1}) = r_{q}(m_{2}), that is, the story m_{1} starts where m_{2} ends. The inverse of σ_{p,q} combines a pair m_{1} ∈ M_{p}, m_{2} ∈ M_{q} of stories of lengths p and q to a story m_{1} ∘ m_{2} of length pq, provided m_{1} starts where m_{2} ends. The assumption that σ_{p,q} be a homeomorphism says that m ∈ M_{pq} and (m_{1}, m_{2}) ∈ M_{p} × M_{q} with s_{p}(m_{1}) = r_{q}(m_{2}) determine each other uniquely and continuously. The length 1 ∈ P is the neutral element, so nothing can happen in time 1, and adding a story of length 1 before or after another story does nothing. This means that M_{1} = X and that σ_{p,1} and σ_{1,q} are the canonical maps. The associativity conditions (4.7) say that the two ways of cutting a story of length pqt into three pieces of length p, q and t give the same results. If P is a free monoid on n generators (which, however, is not Ore), then the situation above may be interpreted as describing a game where the players may do n different things in each time interval. If, say, the player has the three options a, b, c, then p = baac means a time interval of length 4 in which the player first does c, then twice a, then b. If the game was in situation x ∈ X initially, then the points in s_{p}^{−1} (x) ⊆ M_{p} are the possible game developments in this length-4 time period, provided the player’s actions are baac. And r_{p}(m) for m ∈ s_{p}^{−1} (x) is the situation after this time period. If s_{p}^{−1} (x) has more than one point, then the game contains randomness. It makes sense to quantify this randomness by a transfer operator with ∑ _{s_{p} (m)=y} μ_{p}(m) = 1 for all x ∈ X, where μ_{p}(m) is the probability that the game develops as in story m, given the initial situation y. We do not add such probabilities to our setup because they are irrelevant for us by Proposition 4.1. A relation in the monoid P means that certain actions of the player always and automatically have the same effect on the game. For instance, if P is the free Abelian monoid ℕ^{n} on n generators, then the order in which the player does various things does not matter. I know no game with this property; so the interpretation through games works best for free monoids. There are three simple cases of actions by proper topological correspondences:

(1) M_{p} = X and s_{p} = id_{X} for all x ∈ X, p ∈ P; that is, a situation x ∈ X determines its future uniquely;

(2) M_{p} = X and r_{p} = id_{X} for all x ∈ X, p ∈ P; that is, a situation x ∈ X determines its past uniquely;

(3) M_{p} is arbitrary, but s_{p} = r_{p} for all p ∈ P; that is, the situation never changes; then s_{p} = r_{p} must be both proper and a local homeomorphism; equivalently, it is a finite covering map.

From now on, we assume that P is a right Ore monoid. In this case, the Cuntz–Pimsner algebra of the product system (E_{p}, μ_{p,q}) over P is described more concretely in Section 3. We are going to identify this Cuntz–Pimsner algebra with the groupoid C^{∗}-algebra of an étale, locally compact groupoid H. We first describe the object space H^{0} of this groupoid. The first associativity condition in (4.7) says that the spaces M_{p} for p ∈ P and the continuous maps r_{p,q} for p, q ∈ P form a projective system of locally compact spaces indexed by the directed category C_{P}. We let

H^{0} := \underleftarrow{lim}_{C_{P}}(M_{p}, r_{p,q}).

Thus a point in H^{0} consists of m_{p} ∈ M_{p} for all p ∈ P that satisfy r_{p,q}(m_{pq}) = m_{p} for all p, q ∈ P. In other words, the m_{p} are stories that are consistent in the sense that m_{p} is the ending of m_{pq} for each p, q ∈ P. We call a point in H^{0} a complete history and think of m_{p} as describing what happened in the last length-p time period.

Lemma 4.9. The space H^{0} is locally compact and Hausdorff. The projection maps π_{q}: H^{0} → M_{q}, (m_{p})_{p∈P} ↦ m_{q}, are proper for all q ∈ P.

Proof. Fix (m_{p})_{p∈P} ∈ H^{0} and let K ⊂ X = M_{1} be a compact neighbourhood of m_{1}. The preimage of K in H^{0} is the subset of all (m′_{p})_{p∈P} with m′_{1} ∈ K and hence r_{p}(m′_{p}) ∈ K for all p ∈ P. Since all the maps r_{p} are proper, the subsets r_{p}^{−1} (K) ⊆ M_{p} are compact. Hence so is the product L := ∏ _{p∈I} r_{p}^{−1} (K) by Tychonov’s Theorem. Thus the map π_{1} : H^{0} → X is proper. The same argument shows that all the maps π_{q} are proper. Since L is also a compact neighbourhood of (m_{p})_{p∈P} in H^{0}, the space H^{0} is locally compact. If (m′_{p}) ≠ (m_{p}), then there is p ∈ P with m′_{p} ≠ m_{p}. There are open neighbourhoods in M_{p} that separate m′_{p} and m_{p}. These yield open neighbourhoods in H^{0} that separate (m′_{p}) and (m_{p}), so H^{0} is Hausdorff. □

Given a complete history (m_{p})_{p∈P} and t ∈ P, we may forget what happened in the last time period of length t; this gives another complete history, defined formally by m′_{p} := s_{t,p}(m_{tp}) for p ∈ P; the second condition in (4.7) implies r_{p,q}(m′_{pq}) = m′_{p} for all p, t ∈ P, that is, (m′_{p})_{p∈P} is again a complete history as expected. Thus (m_{p}) ↦ (m′_{p}) defines a map s̃_{t} : H^{0} → H^{0}.

Lemma 4.10. Let t ∈ P. The map (π_{t}, s̃_{t}): H^{0} → M_{t} ×_{s_{t},X,π_{1}} H^{0} is a homeomorphism, and s̃_{t} : H^{0} → H^{0} is a local homeomorphism. If s_{t} is surjective, so is s̃_{t}. If s_{t} is a homeomorphism, so is s̃_{t}.

Proof. Let (m_{p})_{p∈P} ∈ H^{0}. Then π_{t}(m_{p}) = m_{t} and s̃_{t}((m_{p})) = (s_{t,p}(m_{tp}))_{p∈P} . Since m_{tp} = r_{t,p}(m_{tp}) · s_{t,p}(m_{tp}) = m_{t} · s̃_{t}((m_{p}))_{p}, we have π_{1} ∘ s̃_{t} = s_{t} ∘ π_{t}, that is, the image of (π_{t}, s̃_{t}) is contained in the fibre product M_{t} ×_{s_{t},X,π_{1}} H^{0}. Since the map (π_{t}, s̃_{t}) is clearly continuous, we must prove that it is a bijection with a continuous inverse. So we let (m′_{p}) ∈ H^{0} be a complete history and let m_{t} ∈ M_{t} be a length-t story with m′_{1} = s_{t}(m_{t}) ∈ X. We must show that there is a unique complete history (m_{p}) with given m_{t} and with s̃_{t}((m_{p})) = (m′_{p}), which depends continuously on (m′_{p}) and m_{t}. First assume that (m_{p})_{p∈P} as above has been found. Let q ∈ P. Then we may write qu = tp for some u, p ∈ P because P is a right Ore monoid. The story m_{tp} is the concatenation m_{t}∘m′_{p} of r_{t,p}(m_{pt}) = m_{t} and s_{t,p}(m_{tp}) = m′_{p}, which exists because r_{p}(m′_{p}) = m′_{1} = s_{t}(m_{t}). Thus m_{q} = r_{q,u}(m_{qu}) = r_{q,u}(m_{t} ∘ m′_{p}). So there is at most one possible solution (m_{p})_{p∈P} , and it depends continuously on (m′_{p}) and m_{t}. We must show that the length-q stories m_{q} := r_{q,u}(m_{t} ∘ m′_{p}) with u, p ∈ P as above form a complete history, that is, r_{q,v}(m_{qv}) = m_{q} for all q, v ∈ P. We have m_{qv} = r_{qv,u_{2}} (m_{t} ∘ m′_{p_{2}} ) for some u_{2}, p_{2} ∈ P with qvu_{2} = tp_{2}. Since P is a right Ore monoid, there are u_{3}, u_{4} ∈ P with vu_{2}u_{3} = uu_{4}. Then tp_{2}u_{3} = qvu_{2}u_{3} = quu_{4} = tpu_{4}. Since P is a right Ore monoid, there is u_{5} ∈ P with p_{2}u_{3}u_{5} = pu_{4}u_{5}. To simplify notation, we replace (u_{3}, u_{4}) by (u_{3}u_{5}, u_{4}u_{5}); thus p_{2}u_{3} = pu_{4}. Since quu_{4} = tpu_{4}, we could also use (uu_{4}, pu_{4}) instead of (u, p) to define m_{q}. The associativity conditions in (4.7) show that this gives the same result:

r_{q,uu_{4}} (m_{t} ∘ m′_{pu_{4}} ) = r_{q,u}r_{qu,u_{4}} (m_{t} ∘ m′_{pu_{4}} )

= r_{q,u}r_{qu,u_{4}} (m_{t} ∘ m′_{p} ∘ s_{p,u_{4}} (m′_{pu_{4}} )) = r_{q,u}(m_{t} ∘ m′_{p}).

Similarly, we get the same result for m_{qv} if we use (u_{2}u_{3}, p_{2}u_{3}) instead of (u_{2}, p_{2}). Thus we may assume that p_{2} = p and u = vu_{2}. Then

r_{q,v}(m_{qv}) = r_{q,v}(r_{qv,u_{2}} (m_{t} ∘ m′_{p})) = r_{q,vu_{2}} (m_{t} ∘ m′_{p}) = r_{q,u}(m_{t} ∘ m′_{p}) = m_{q}.

This finishes the proof that (π_{t}, s̃_{t}) is a homeomorphism. This homeomorphism transforms the map s̃_{t} into the second coordinate projection M_{t} ×_{s_{t},X,π_{1}} H^{0} → H^{0}, which is a (local) homeomorphism if s_{t} is one, and surjective if s_{t} is. □

First forgetting the last length-t time period and then the last length-u time period gives the same result as directly forgetting the last time period of length tu. That is, s̃_{u} ∘ s̃_{t} = s̃_{tu} for all t, u ∈ P. Formally, this follows from the third condition in (4.7). Thus the monoid P^{op} acts on H^{0} by local homeomorphisms. Why do we get the opposite monoid here? The maps s̃_{t} : H^{0} → H^{0} and r̃_{t} := id_{H^{0}} form an action of P by topological correspondences with the extra property that any situation determines its past uniquely: a “situation” in H^{0} is a complete history, which simply contains its past. Thus we still have an action by topological correspondences, but one where the maps r̃_{p} are all identity maps, so that we may forget about them. This gives an action of the opposite monoid P^{op} by local homeomorphisms because of the direction of the maps s̃_{p}.

Example 4.11. Suppose that we start with an action of P on X by proper maps r_{p} and let s_{p} be identity maps; that is, every situation determines its future. Then the maps s̃_{t} are homeomorphisms by Lemma 4.10. Thus our action of P^{op} on H^{0} extends to the group completion G of P^{op}. The groupoid model we are going to construct is the transformation groupoid of this group action.

Definition 4.12. The transformation groupoid H := P^{op} ⋉ H^{0} associated to the P^{op}-action (s̃_{p}) on H^{0} by local homeomorphisms has object space H^{0}, arrow set

H^{1} := {(x, g, y) ∈ H^{0} × G × H^{0} | ∃p_{1}, p_{2} ∈ P, g = p_{1}p_{2}^{−1} , s̃_{p_{1}} (x) = s̃_{p_{2}} (y)},

range and source maps r(x, g, y) := x, s(x, g, y) := y, and multiplication

(x_{1}, g_{1}, y_{1}) · (x_{2}, g_{2}, y_{2}) := (x_{1}, g_{1}g_{2}, y_{2})

if y_{1} = x_{2}. The unit on x ∈ H^{0} is (x, 1, x), the inverse of (x, g, y) is (y, g^{−1} , x). We describe the topology on H^{1}. For p_{1}, p_{1} ∈ P, let

H_{p_{1} ,p_{2}}^{1} := H^{0} ×_{s̃_{p_{1}} ,H^{0} ,s̃_{p_{2}}} H^{0} := {(x, y) ∈ H^{0} × H^{0} | s̃_{p_{1}} (x) = s̃_{p_{2}} (y)},

the fibre product of the diagram H^{0} ⟶^{s̃_{p_{1}}} H^{0} ⟵^{s̃_{p_{2}}} H^{0}. We give each H_{p_{1} ,p_{2}}^{1} the subspace topology from the product H^{0} × H^{0}, and ⨆ _{p_{1} ,p_{2} ∈P} H_{p_{1} ,p_{2}}^{1} the disjoint union topology. We map H_{p_{1} ,p_{2}}^{1} → H^{1} by (x, y) ↦ (x, p_{1}p_{2}^{−1} , y) ∈ H^{1}. This gives a surjection ⨆ H_{p_{1} ,p_{2}}^{1} → H^{1}. We give H^{1} the quotient topology from this map.

To verify that the transformation groupoid has desirable properties, we rewrite it using filtered colimits. Let H_{g}^{1} := {(x, g, y) ∈ H^{1}} for g ∈ G; so H^{1} = ⨆ _{g∈G} H_{g}^{1}. We describe H_{g}^{1} for fixed g ∈ G as a colimit over C_{P}^{g} (see Definition 3.14), which is a filtered category by Lemma 3.15. If p_{1}q = p_{3}, p_{2}q = p_{4}, then s̃_{p_{1}} (x) = s̃_{p_{2}} (y) implies s̃_{p_{3}} (x) = s̃_{p_{4}} (y), so H_{p_{1} ,p_{2}}^{1} ⊆ H_{p_{3} ,p_{4}}^{1} ⊆ H^{0} × H^{0}.

Since right multiplication with q is locally injective, any (x, y) ∈ H_{p_{3} ,p_{4}}^{1} has a neighbourhood in H^{0} × H^{0} so that for (x′, y′) in this neighbourhood, s̃_{p_{1}} (x′) ≠ s̃_{p_{2}} (y′) implies s̃_{p_{1} q}(x′) ≠ s̃_{p_{2} q}(y′). Thus the subset H_{p_{1} ,p_{2}}^{1} is relatively open in H_{p_{3} ,p_{4}}^{1} ; so the spaces H_{p_{1} ,p_{2}}^{1} for (p_{1}, p_{2}) ∈ P^{2} and the inclusion maps H_{p_{1} ,p_{2}}^{1} → H_{p_{3} ,p_{4}}^{1} form a diagram of subsets of H^{0}×H^{0} with open inclusion maps.

Lemma 4.13. If p_{1}p_{2}^{−1} = p_{3}p_{4}^{−1} , then there are p_{5}, p_{6} ∈ P with p_{5}p_{6}^{−1} = p_{1}p_{2}^{−1} and so that both H_{p_{1} ,p_{2}}^{1} and H_{p_{3} ,p_{4}}^{1} are open subsets of H_{p_{5} ,p_{6}}^{1} ; hence the subspace topologies from H_{p_{1} ,p_{2}}^{1} and H_{p_{3} ,p_{4}}^{1} coincide on H_{p_{1} ,p_{2}}^{1} ∩H_{p_{3} ,p_{4}}^{1} , and this subset is open both in H_{p_{1} ,p_{2}}^{1} and in H_{p_{3} ,p_{4}}^{1} . Each H_{p_{1} ,p_{2}}^{1} is open in H_{g}^{1}, and the topology on H_{g}^{1} restricts to the given topology on each H_{p_{1} ,p_{2}}^{1} .

Proof. Since C_{P}^{g} is filtered, there is an object (p_{5}, p_{6}) that dominates both (p_{1}, p_{2}) and (p_{3}, p_{4}). This has all required properties. Thus all the embeddings H_{p_{1} ,p_{2}}^{1} → H_{p_{3} ,p_{4}}^{1} are open. This implies that the quotient topology on H_{g}^{1} = ⋃ _{(p_{1} ,p_{2} )∈R_{g}} H_{p_{1} ,p_{2}}^{1} from ⨆ _{(p_{1} ,p_{2} )∈R_{g}} H_{p_{1} ,p_{2}}^{1} restricts to the given topology on each subset H_{p_{1} ,p_{2}}^{1} . □

Thus the subsets H_{p_{1} ,p_{2}}^{1} for p_{1}, p_{2} ∈ P form an open covering of H^{1}, and the topology on H^{1} restricts to the usual topology on each H_{p_{1} ,p_{2}}^{1} . In the following, we identify H_{p_{1} ,p_{2}}^{1} with its image in H^{1}, which is an open subset.

Proposition 4.14. The groupoid H is étale, locally compact and Hausdorff. The decomposition H^{1} = ⨆ _{g∈G} H_{g}^{1} satisfies H_{g}^{1}·H_{h}^{1} ⊆ H_{gh}^{1} and (H_{g}^{1})^{−1} = H_{g^{−1}}^{1} . If the maps s_{p} for p ∈ P are surjective, then H_{g}^{1} · H_{h}^{1} = H_{gh}^{1} for all g, h ∈ G. The groupoid H_{1}^{1} ⊆ H^{1} is an increasing union of open subgroupoids that are proper and étale and describe equivalence relations on H^{0}.

Proof. The space H^{0} is locally compact and Hausdorff by Lemma 4.9. The coordinate projections H_{p_{1} ,p_{2}}^{1} ⇉ H^{0} are étale because P acts by local homeomorphisms. Since the subsets H_{p_{1} ,p_{2}}^{1} for p_{1}, p_{2} ∈ P form an open covering of H^{1}, the coordinate projections H^{1} ⇉ H^{0} are étale. Any two points of H_{g}^{1} are contained in the same Hausdorff, locally compact, open subset H_{p_{1} ,p_{2}}^{1} for suitable p_{1}, p_{2}, so they may be separated by open subsets of H_{g}^{1}; since the subsets H_{g}^{1} are open, it is also possible to separate points in H_{g}^{1} and H_{h}^{1} for g ≠ h. Thus H^{1} is Hausdorff. If the maps s_{p} for p ∈ P are surjective, then so are the maps s̃_{p} for p ∈ P by Lemma 4.10. Now let (x, gh, y) ∈ H_{gh}^{1}. Hence there are p_{1}, p_{2} ∈ P with gh = p_{1}p_{2}^{−1} and s̃_{p_{1}} (x) = s̃_{p_{2}} (y). Write g = p_{3}p_{4}^{−1} , h = p_{5}p_{6}^{−1} . Then also g = p_{3}q(p_{4}q)^{−1} and h = (p_{5}t)(p_{6}t)^{−1} for all q, t ∈ P. The Ore condition (O1) allows us to choose q and t such that p_{4}q = p_{5}t. Hence we may assume without loss of generality that p_{4} = p_{5}. Then

p_{1}p_{2}^{−1} = gh = p_{3}p_{4}^{−1} p_{5}p_{6}^{−1} = p_{3}p_{6}^{−1} .

If s̃_{p_{1}} (x) = s̃_{p_{2}} (y), then also s̃_{p_{1} t}(x) = s̃_{p_{2} t}(y) for any t ∈ P, so we may rewrite gh = (p_{1}t)(p_{2}t)^{−1} . We may also replace (p_{3}, p_{4}, p_{5}, p_{6}) by (p_{3}q, p_{4}q, p_{5}q, p_{6}q) for any q ∈ P. Choosing q and t by condition (O1), we may achieve p_{3}q = p_{1}t, p_{6}q = p_{2}t, by the definition of the group G. Hence we find p_{1}, p_{2}, p_{3} ∈ P with gh = p_{1}p_{2}^{−1} , g = p_{1}p_{3}^{−1} , h = p_{3}p_{2}^{−1} and s̃_{p_{1}} (x) = s̃_{p_{2}} (y). Since s̃_{p_{3}} is surjective, we may choose z ∈ H^{0} with s̃_{p_{3}} (z) = s̃_{p_{1}} (x) = s̃_{p_{2}} (y). Then (x, g, z) ∈ H_{g}^{1} and (z, h, y) ∈ H_{h}^{1} satisfy (x, g, z) · (z, h, y) = (x, gh, y). The open subgroupoid H_{1}^{1} is defined as the union of H_{p_{1} ,p_{2}}^{1} with (p_{1}, p_{2}) ∈ R_{1}. Since C_{P} is cofinal in C_{P}^{1} by Lemma 3.15, H_{1}^{1} = ⋃ _{p∈P} H_{p,p}^{1}. Here H_{p,p}^{1} is the set of all (x, y) ∈ H^{0} ×H^{0} with s̃_{p}(x) = s̃_{p}(y), and it carries the subspace topology from H^{0} × H^{0}. So H_{p,p}^{1} is a proper equivalence relation on H^{0}, and H_{1}^{1} is the union of these open subgroupoids. □

If P is countable, then we may choose a cofinal sequence in C_{P}^{1} and write H_{1}^{1} as an increasing union of a sequence of proper étale equivalence relations. Hence H_{1}^{1} is an approximately proper equivalence relation in the notation of [71]. These are called hyperfinite relations in [47]. We allow ourselves to call H_{1}^{1} approximately proper also if P is uncountable, replacing a sequence of proper (finite) open subrelations by a directed set of such subrelations. If the maps s_{p} for p ∈ P are surjective, then the subsets H_{g}^{1} for g ∈ G form a G-grading in the notation of [11]. This is equivalent to an action of G on the groupoid H_{1}^{1} by Morita equivalences with transformation groupoid H. Thus we may think of H as the transformation groupoid associated to an action of G on the noncommutative orbit space H^{1}/H_{1}^{1}. Points in this orbit space are equivalence classes of complete histories, where two complete histories are identified if they coincide in the distant past, that is, s̃_{p}(x) = s̃_{p}(y) for some p ∈ P. The group G acts on this by “time translations.” If the maps s_{p} are not surjective, then the G-action on H^{1} is only a partial action because time translations x ↦ px into the future are not everywhere defined. A partial G-action is the same as an action of a certain inverse semigroup associated to G, see [23].

Definition 4.15. A situation x ∈ X is (historically) possible if x ∈ r_{p}(M_{p}) for all p ∈ P.

Let X′ ⊆ X be the subset of possible situations. We have X′ = X if and only if all the maps r_{p} are surjective. Let M′_{p} = s_{p}^{−1} (X′) and let r′_{p} and s′_{p} be the restrictions of r_{p} and s_{p} to M′_{p}. Any situation that occurs at some point in a complete history is possible, so we have m_{p} ∈ M′_{p} for any (m_{p})_{p∈P} ∈ H^{0}. Conversely, a situation that is possible is the endpoint m_{1} of some complete history (m_{p})_{p∈P} because the maps r_{p} are proper (Lemma 4.9) and a projective limit of non-empty compact spaces is non-empty. Thus π_{p}^{−1} (H^{0}) = M′_{p}, r′_{p}(M′_{p}) = X′ for all p ∈ P, and s′_{p}(M′_{p}) ⊆ X′ by associativity: if a situation has a possible past, then it is itself possible because we may concatenate stories. We still have isomorphisms M′_{p} ×_{X′} M′_{q} ≅ M′_{pq}, so restricting to the possible situations gives a new action by topological correspondences. By construction, both systems (X, M, s_{p}, r_{p}, σ_{p,q}) and (X′, M′_{p}, s′_{p}, r′_{p}, σ′_{p,q}) have the same complete histories and thus the same transformation groupoid H.

Lemma 4.16. Let q, a_{1}, a_{2} ∈ P satisfy qa_{1} = qa_{2}. Then r_{q,a_{1}} |_{M′_{qa_{1}}} = r_{q,a_{2}} |_{M′_{qa_{2}}} .

Proof. Condition (O2) gives us b ∈ P with a_{1}b = a_{2}b. The associativity property (4.7) of the maps r_{p,p} gives r_{q,a_{i}} ∘ r_{qa_{i} ,b} = r_{q,a_{i} b}. Hence r_{q,a_{1}} and r_{q,a_{2}} coincide on the range of r_{q,a_{1} b} = r_{q,a_{2} b}. The subspace M′_{qa_{1}} is contained in that range. □

Write p ≥ q for p, q ∈ P if there is a ∈ P with p = qa. Lemma 4.16 shows that after restricting to the possible situations, the truncation map r_{q,a} : M′_{p} → M′_{q} for p ≥ q does not depend on the choice of a.

Theorem 4.17. The groupoid C^{∗}-algebra C^{∗}(H) is canonically isomorphic to the Cuntz–Pimsner algebra of the product system (E_{p})_{p∈P} over P described above.

Proof. The proof has two parts. In the first part, we show that the original action on X has the same colimit as the induced action on H^{0}. So we are reduced to the special case of actions by correspondences of the special form where the maps r_{p}: M_{p} → X are all identity maps. In the second part, we do this case by hand. Let D be a C^{∗}-algebra. A transformation from the product system (E_{p})_{p∈P} to D consists of a correspondence F to D together with isomorphisms of correspondences V_{p}: E_{p} ⊗_{C_{0} (X)} F ≅ F. The left action of C_{0}(X) on E_{p} extends to an action of C_{0}(M_{p}) by pointwise multiplication. Thus we get a canonical left action of C_{0}(M_{p}) on E_{p} ⊗_{C_{0} (X)} F ≅ F, where we use use the isomorphism V_{p}. Thus C_{0}(M_{p}) acts on F in a canonical way for each p ∈ P. If p, q ∈ P, then the isomorphism V_{pq}: E_{pq} ⊗_{C_{0} (X)} F ≅ F is equal to the composite isomorphism where we first identify E_{pq} ≅ E_{p} ⊗_{C_{0} (X)} E_{q} and then apply V_{q} and V_{p}. As a consequence, the action of C_{0}(M_{p}) on F is the composite of the action of C_{0}(M_{pq}) on F and the ^{∗}-homomorphism r_{p,q}: C_{0}(M_{p}) → C_{0}(M_{pq}). So the left actions of C_{0}(M_{p}) fit together to an action of the inductive limit

\underrightarrow{lim}_{C_{P}}(C_{0}(M_{p}), r_{p,q}^{∗}) ≅ C_{0}( \underleftarrow{lim}_{C_{P}}(M_{p}, r_{p,q})) = C_{0}(H^{0}).

Thus F carries a nondegenerate ^{∗}-representation of C_{0}(H^{0}), turning it into a correspondence F̃ from C_{0}(H^{0}) to D. Now let Ẽ_{p}:= E_{p} ⊗_{C_{0} (X)} C_{0}(H^{0}). Then

Ẽ_{p}⊗_{C_{0} (H^{0} )} F̃ ≅ E_{p} ⊗_{C_{0} (X)} C_{0}(H^{0}) ⊗_{C_{0} (H^{0} )} F̃ ≅ E_{p} ⊗_{C_{0} (X)} F ≅ F,

where the last isomorphism is V_{p} and all other isomorphisms are trivial. By definition, the correspondence Ẽ_{p} from C_{0}(X) to C_{0}(H^{0}) is obtained from a topological correspondence as well, namely, we replace M_{p} by M̃_{p} = M_{p}×_{s_{p} ,X,π_{1}} H^{0} and use the map (m, ω) ↦ r_{p}(m) as range and the map (m, ω) ↦ ω as source map. Lemma 4.10 describes a homeomorphism M̃_{p} ≅ H^{0} such that the second coordinate projection becomes s̃_{p} : H^{0} → H^{0}. We let r̃_{p} be the identity map H^{0} → H^{0} and thus turn Ẽ_{p} into a correspondence from C_{0}(H^{0}) to itself. Since s̃_{p} ∘s̃_{q} = s̃_{qp}, the usual composition of topological correspondences defines an action of P on H^{0} by topological correspondences. Thus the associated C^{∗}-correspondences Ẽ_{p} form a product system over P with unit fibre C_{0}(H^{0}). We claim that the isomorphism of Hilbert modules Ẽ_{p} ⊗_{C_{0} (H^{0} )} F̃ ≅ F̃ constructed above is an isomorphism of correspondences from C_{0}(H^{0}) to D with this choice of left action of C_{0}(H^{0}) on Ẽ_{p}. It suffices that C_{0}(M_{q}) acts in the same way on both sides for each q ∈ P. The Ore condition (O1) gives t, u ∈ P with pt = qu. Since the action of C_{0}(M_{q}) factors through C_{0}(M_{qu}) and qu = pt, we may assume q = pt. The left action of C_{0}(M_{pt}) on Ẽ_{p}⊗_{C_{0} (H^{0} )} F̃ is obtained as follows. First, as Hilbert D-modules, we identify

Ẽ_{p}⊗_{C_{0} (H^{0} )} F̃ ≅ (E_{p} ⊗_{C_{0} (X)} C_{0}(M_{t})) ⊗_{C_{0} (M_{t})} (E_{t} ⊗_{C_{0} (X)} F).

Then we identify E_{p} ⊗_{C_{0} (X)} C_{0}(M_{t}) with the C^{∗}-correspondence from C_{0}(M_{pt}) to C_{0}(M_{t}) associated to the topological correspondence M_{pt} ≅ M_{p} ×_{s_{p} ,X,r_{t}} M_{t} → M_{t}, where the range map is the homeomorphism σ_{p,t} and the source map is pr_{2}. By Lemma 4.3, the composite of this with the C^{∗}-correspondence E_{t} from C_{0}(M_{t}) to C_{0}(X) is the C^{∗}-correspondence associated to the composite topological correspondence, (M_{p}×_{s_{p} ,X,r_{t}} M_{t})×_{pr_{2} ,M_{t},r_{t}} M_{t}. This is M_{p}×_{s_{p} ,X,r_{t}} M_{t} now viewed as a topological correspondence from M_{pt} to X. As such, it is isomorphic to M_{pt}. Thus we may also get the left action of C_{0}(M_{pt}) on Ẽ_{p}⊗_{C_{0} (H^{0} )} F̃ by identifying this in the canonical way with E_{pt} ⊗_{C_{0} (X)} F and then acting on the first tensor factor by pointwise multiplication. This, however, is exactly how the left action of C_{0}(M_{pt}) on F̃ is defined. Thus the left actions of C_{0}(M_{t}) on Ẽ_{p}⊗_{C_{0} (H^{0} )} F̃ and F̃ coincide as expected. Since C_{0}(X) ⊆ C_{0}(H^{0}), we may view E_{p} as a subspace of Ẽ_{p}. The map from the algebraic tensor product E_{p} ⊙ F to Ẽ_{p}⊗_{C_{0} (H^{0} )} F has dense range because the target is isomorphic to F, which is also isomorphic to the correspondence tensor product E_{p} ⊗_{C_{0} (X)} F, and there E_{p} ⊙ F is certainly dense. Hence the map from E_{p} ⊙ E_{q} ⊙ F to Ẽ_{p}⊗_{C_{0} (H^{0} )} Ẽ_{q}⊗_{C_{0} (H^{0} )} F also has dense range. The coherence condition for a transformation of product systems (3.2) holds on this dense subspace by assumption, and hence it holds everywhere. Thus our isomorphisms of C^{∗}-correspondences Ẽ_{p}⊗_{C_{0} (H^{0} )} F ≅ F form a transformation as expected. Thus a transformation from the product system E_{p} to D gives a transformation from the product system Ẽ_{p} to D, with the same underlying Hilbert module F. Conversely, a transformation from the product system Ẽ_{p} to D gives one from the product system E_{p} to D because E_{p} ⊆ Ẽ_{p}. Since E_{p}⊙F is dense in Ẽ_{p}⊗_{C_{0} (H^{0} )} F, these two constructions must be inverse to each other. Summing up, we have found a bijection between the transformations from our two product systems to D that does not change the underlying Hilbert module F. The results in Section 2 show that such transformations are in bijection with nondegenerate representations of the Cuntz–Pimsner algebras of the two product systems on F, respectively. Having found bijections between the representations of both Cuntz–Pimsner algebras on any Hilbert module, we conclude by the Yoneda Lemma that they must be isomorphic. We have now reduced the general case of an action by topological correspondences to the special case of an action with M_{p} = X and r_{p} = id for all p ∈ P. This case is much easier because the groupoid model has H^{0} = X, so we merely take a transformation groupoid and do not change the underlying object space. By definition, C^{∗}(H) is the C^{∗}-completion of the dense ^{∗}-subalgebra C_{c}(H^{1}) of compactly supported, continuous functions on H^{1}, equipped with the usual convolution and involution

f_{1} ∗ f_{2}(h) := ∑_{h_{1} h_{2} =h} f_{1}(h_{1})f_{2}(h_{2}), f^{∗}(h) := \overline{f(h^{−1} )} ,

for f_{1}, f_{2}, f ∈ C_{c}(H^{1}), h ∈ H^{1} (see [28, Section 3]). Here “C^{∗}-completion” means that we complete in the largest C^{∗}-seminorm on C_{c}(H^{1}). There is no need to assume boundedness for the I-norm. First, the argument in [28] shows that every Hilbert space representation and hence every C^{∗}-seminorm is continuous for the inductive limit topology; secondly, [70, Corollaire 4.8] shows that such representations and C^{∗}-seminorms are bounded for the I-norm. The disjoint decomposition H^{1} = ⨆ _{g∈G} H_{g}^{1} gives C_{c}(H^{1}) = ⨁ _{g∈G} C_{c}(H_{g}^{1}). This is a non-saturated G-grading, that is, C_{c}(H_{g}^{1}) ∗ C_{c}(H_{h}^{1}) ⊆ C_{c}(H_{gh}^{1}) and C_{c}(H_{g}^{1})^{∗} = C_{c}(H_{g^{−1}}^{1} ). This G-grading turns C^{∗}(H) into the section algebra of a Fell bundle over G. Of course, our proof will show that this Fell bundle structure corresponds to the same structure on the Cuntz–Pimsner algebra. The space H_{g}^{1} is an increasing union of the open subsets H_{p_{1} ,p_{2}}^{1} . Thus any function in C_{c}(H_{g}^{1}) already belongs to C_{c}(H_{p,q}^{1}) for some p, q ∈ P with pq^{−1} = g:

C_{c}(H_{g}^{1}) = ⋃_{pq^{−1} =g} C_{c}(H_{p,q}^{1}).

Since X = H^{1}, we have H_{p,q}^{1} ≅ X×_{s_{p} ,X,s_{q}} X, the set of pairs (x, y) with s_{p}(x) = s_{q}(x). We are now going to relate C_{c}(X ×_{s_{p} ,X,s_{q}} X) to the space 𝕂(E_{q}, E_{p}) in the description of the Cuntz–Pimsner algebra in the proof of Theorem 3.16. Given a function k ∈ C_{c}(X ×_{s_{p} ,X,s_{q}} X), we define

T_{k}: E_{q} → E_{p},  (T_{k}ξ)(m_{1}) := ∑_{s_{p} (m_{1} )=s_{q}(m_{2} )} k(m_{1}, m_{2})ξ(m_{2});

these sums are uniformly finite for m_{1} in a compact subset because s_{p} and s_{q} are local homeomorphisms and the support of k is compact. The operator T_{k} is a rank-one operator if k(m_{1}, m_{2}) = k_{1}(m_{1})·k_{2}(m_{2}) with k_{1} ∈ C_{c}(X), k_{2} ∈ C_{c}(X). Since functions k of this form are dense in C_{c}(X ×_{s_{p} ,X,s_{q}} X) and the map k ↦ T_{k} is continuous, we have T_{k} ∈ 𝕂(E_{q}, E_{p}) for all k ∈ C_{c}(X ×_{s_{p} ,X,s_{q}} X). Since compactly supported functions are dense in E_{p} and E_{q}, operators of the form T_{k} for k ∈ C_{c}(X ×_{s_{p} ,X,s_{q}} X) are dense in 𝕂(E_{q}, E_{p}). If T_{k} = 0, then k = 0.

We may express the product and involution on compact operators through kernel functions: if k ∈ C_{c}(X ×_{s_{p} ,X,s_{q}} X) and l ∈ C_{c}(X ×_{s_{q},X,s_{t}} X), then T_{k} ∘ T_{l} has the kernel (m_{1}, m_{2}) ↦ ∑ _{s_{q}(m)=s_{p} (m_{1} )=s_{t}(m_{2} )} k(m_{1}, m)l(m, m_{2}), and the adjoint T_{k}^{∗} has the kernel (m_{1}, m_{2}) ↦ \overline{k(m_{2}, m_{1})}. These formulas correspond to the convolution and involution in C_{c}(H^{1}). Therefore, the map k ↦ T_{k} is an injective ^{∗}-homomorphism with dense range from C_{c}(H^{1}) = ∑ _{p,q∈P} C_{c}(H_{p,q}^{1}) to the ^{∗}-algebra ∑ _{g∈G} O_{g} of compactly supported sections of the Fell bundle (O_{g})_{g∈G} . It remains to show that this ^{∗}-homomorphism extends to an isomorphism between the C^{∗}-completions. It suffices to prove that the restriction of any ∗-representation of C_{c}(H^{1}) to C_{c}(H_{g}^{1}) is bounded with respect to the norm of O_{g}. Since ∥ξ∥^{2} = ∥ξ^{∗}ξ∥_{O_{1}} for all ξ ∈ O_{g}, this holds for all g once it holds for g = 1. Thus it remains to show that the unit fibre O_{1} of the Cuntz–Pimsner algebra is the C^{∗}-completion of C_{c}(H_{1}^{1}) for the subgroupoid H_{1}^{1}. If p ∈ P, then the subset H_{p,p}^{1} = X ×_{s_{p} ,X,s_{p}} X of H_{1}^{1} is the groupoid describing the equivalence relation ∼_{p} induced by the map s_{p}. This equivalence relation is proper, that is, the map X → X/∼_{p} is proper, since s_{p} is a local homeomorphism. Hence the C^{∗}-algebra of the groupoid H_{p,p}^{1} is Morita–Rieffel equivalent to C_{0}(s_{p}(X)). The Hilbert bimodule constructed in the proof of this Morita–Rieffel equivalent in [61] is exactly our correspondence E_{p}. Hence C^{∗}(H_{p,p}^{1}) = 𝕂(E_{p}). The C^{∗}-algebras C^{∗}(H^{1}) and O_{1} are the colimits of the diagrams of C^{∗}-algebras C^{∗}(H_{p,p}^{1}) and 𝕂(E_{p}) over the filtered category C_{P}. Hence they are also canonically isomorphic. □

5. Some relations to previous work

The construction of groupoid models above is very general and contains many known constructions. We discuss some of them in this section. First let P = (ℕ, +). An action of ℕ by topological correspondences is already determined by the single topological correspondence (M_{1}, r_{1}, s_{1}), where s_{1} must be a local homeomorphism (1 ∈ ℕ is not the unit element here, there is a conflict with our usual multiplicative notation). This topological correspondence is the same as a topological graph. We assume r_{1} to be proper to get a proper topological correspondence; then the composite correspondences M_{n} for n ∈ ℕ are automatically proper. We also assume r_{1} to be surjective; equivalently, all maps r_{n} are surjective and X = X′. What we are dealing with is a rowfinite topological graph without sources, which we simply call regular. The space X is its space of vertices, and M_{1} is its space of edges, with m ∈ M_{1} giving an edge from s_{1}(x) to r_{1}(x). A point in M_{n} is a path in the topological graph of length n, and the maps s_{n} and r_{n} send such a path to its initial and final point. The homeomorphism M_{n} ×_{s_{n} ,X,r_{m}} M_{m} ≅ M_{n+m} builds a path of length n + m by concatenating two paths of length n and m. If the space X of vertices is discrete, then so is the space of edges M_{1} because s_{1} is a local homeomorphism. Thus the case where X is discrete gives the ordinary graphs among the topological graphs.

For a regular topological graph, the topological graph C^{∗}-algebra of Katsura [43–45] is, by definition, the same absolute Cuntz–Pimsner algebra that we study. If there are sources, that is, r_{1} is not surjective, then neither our (absolute) Cuntz–Pimsner algebra nor its groupoid model see a difference between the topological graph (X, M_{1}, r_{1}, s_{1}) and its restriction (X′, M′_{1}, r′_{1}, s′_{1}), which now has surjective r′_{1}. Thus we get the topological graph C^{∗}-algebra of the regular graph (X′, M′_{1}, r′_{1}, s′_{1}); this is quite different from Katsura’s C^{∗}-algebra for the original topological graph. Groupoid models played an important role in the definition of (discrete) graph C^{∗}-algebras by Kumjian, Pask, Raeburn and Renault in [48]. They are, however, not used by Katsura to develop the theory of topological graph C^{∗}-algebras. A pretty general construction of a groupoid model for topological graph C^{∗}-algebras is due to Deaconu [19], under the extra assumption that both maps s_{1} and r_{1} be surjective local homeomorphisms and both spaces X and M_{1} be compact. These assumptions are removed by Yeend [78], who constructs groupoid models for the more general class of higher-rank topological graphs. His construction works for all rank-1 graphs, that is, he may allow r_{1} to be any map. The construction simplifies, however, if r_{1} is proper and surjective. Since points in M_{n} are paths of length n, our complete histories in H^{0} are the same as infinite paths in the topological graph. The map π_{n} : H^{0} → M_{n} gives the initial segment of a path of length n. The local homeomorphism s̃_{n} truncates an infinite path by throwing away the initial segment of length n. The groupoid H is ℤ-graded,

H = ⨆_{n∈ℤ} H_{n},

and the unit fibre H_{0} ⊆ H describes the equivalence relation of tail equivalence: we identify two infinite paths if they eventually become equal (without shift). The whole groupoid H combines tail equivalence with the shift map on infinite paths: an arrow in H_{n} means that two infinite paths become eventually equal if we also shift one of them by n steps. The construction above is exactly how the usual groupoid model for a regular topological graph is constructed. Thus our groupoid H is the same as Yeend’s groupoid model for the C^{∗}-algebra of a regular topological graph, which in turn generalises the groupoid models in [19, 48]. In particular, we get the familiar groupoid model for the C^{∗}-algebra of a row-finite graph without sources. In the irregular case, Yeend adds certain finite paths to H^{0}, and he defines the topology on the resulting space of “boundary paths” carefully to get a locally compact space. Next consider P = (ℕ^{k}, +) for some k ≥ 2. An action of the Ore monoid ℕ^{k} by topological correspondences is the same, almost by definition, as a topological rank-k graph. The case where the underlying space X is discrete corresponds to an ordinary rank-k graph. Topological higher-rank graphs are introduced by Yeend, who also describes a groupoid model for them in [79]. He requires the source maps to be local homeomorphisms, but does not require the range maps to be proper; instead, he assumes a weaker condition called “compact alignment,” which may be formulated for lattice-ordered semigroups. He constructs groupoid models for the Toeplitz C^{∗}-algebra and the relative Cuntz–Pimsner algebra of the product system over ℕ^{k} associated to a compactly aligned topological higher-rank graph. The relative and absolute Cuntz–Pimsner algebras agree if and only if all range maps r_{p} are surjective (“no sources”). We call a topological higher-rank graph regular if the maps r_{p} are surjective and proper for all p ∈ ℕ^{k}. In the regular case, the groupoid model constructed by Yeend [79] is the same one that we have constructed above: the boundary paths that form the object space for Yeend’s groupoid model are the same as our complete histories by [79, Lemma 6.6]. In the irregular case, the object space of Yeend’s groupoid combines infinite paths with certain finite and partially infinite paths. It is unclear how to carry this over to actions of Ore monoids. How about groupoid models for actions of semigroups other than ℕ^{k}? Here we are only aware of constructions in particular cases. We discuss two general situations. First, if the action is by local homeomorphisms, then we are already very close to an inverse semigroup action, which is easily translated to an étale groupoid (see [28,31,64]). Secondly, the construction of a semigroup C^{∗}-algebra by Xin Li in [16, 55] may also be based on an action of the semigroup by topological correspondences.

5.1. Semigroups of partial local homeomorphisms. Let P be a monoid, and let P act on a locally compact space X by partial local homeomorphisms, that is, by topological correspondences of the special form

(5.1)  X ⟵−−−−^{inclusion} U_{p} ⟶^{α_{p}} X,

where U_{p} ⊆ X is an open subset and α_{p} is a local homeomorphism. These topological correspondences are only proper if the domains U_{p} are also closed. But the following construction of a groupoid does not need this assumption, and neither does it require the monoid P to be Ore. We do not claim, however, that the groupoid C^{∗}-algebra of the resulting groupoid is isomorphic to the Cuntz–Pimsner algebra of the product system over P associated to our action: our proof only gives this if P is an Ore monoid and the correspondences are proper, that is, the subsets U_{p} are clopen. First we make the multiplication maps explicit. The fibre product U_{p}×_{s_{p} ,X,r_{q}} U_{q} consists of pairs (x, y) with x ∈ U_{p}, y ∈ U_{q}, and α_{p}(x) = y, and the range and source maps on U_{p} ×_{s_{p} ,X,r_{q}} U_{q} take (x, y) to x and α_{q}(y), respectively. Since y = α_{p}(x), the map (x, y) ↦ x identifies U_{p} ×_{s_{p} ,X,r_{q}} U_{q} with U_{p} ∩ α_{p}^{−1} (U_{q}); under this identification, the range and source maps become the inclusion map and the map x ↦ α_{q}(α_{p}(x)), respectively. Thus we must have

(5.2)  U_{pq} = U_{p} ∩ α_{p}^{−1} (U_{q}),  α_{pq} = α_{q} ∘ α_{p}.

Conversely, these conditions give a unique isomorphism of topological correspondences U_{p} ×_{s_{p} ,X,r_{q}} U_{q} ≅ U_{pq}. These isomorphisms automatically verify the associativity condition required for an action of P by topological correspondences. Thus an action of P by topological correspondences of the special form (5.1) is the same as a homomorphism from P^{op} to the monoid of partial local homeomorphisms of X, with the composition of partial local homeomorphisms defined in (5.2). If V ⊆ U_{p} is such that α_{p}|_{V} is injective, then α_{p}|_{V} is a partial homeomorphism on X. Since α_{p} is a local homeomorphism, any point in U_{p} has a neighbourhood V on which α_{p}|_{V} is injective, so that these partial homeomorphisms contained in α_{p} cover α_{p}. Hence we do not lose any information if we replace α_{p} by the set of all partial homeomorphisms α_{p}|_{V} for V ⊆ U_{p} such that α_{p}|_{V} is injective. These partial homeomorphisms α_{p}|_{V} form a semigroup because α_{q}|_{V} ∘ α_{p}|_{W} = α_{pq}|_{W ∩α_{p}^{−1} (V)} and W ∩ α_{p}^{−1} (V) is an open subset of U_{pq} on which α_{pq} is injective by (5.2). We let S be the inverse semigroup of partial homeomorphisms generated by these partial homeomorphisms α_{p}|_{V}. This inverse semigroup acts on X by construction, and this action has an associated transformation groupoid X ⋊S, also called groupoid of germs; see [28, 64]. This groupoid is often the same as the groupoid model constructed in Section 4, but there are some “trivial” counterexamples. The issue is how to define the germ relation. To always get the groupoid constructed in Section 4, we do the following. First, we assume now that P is an Ore monoid with group completion G. Since the construction in Section 4 is only for actions by proper correspondences, we also require that the domains U_{p} are clopen. We let S_{0} be the free inverse semigroup on symbols (p, V) for α_{p}|_{V} as above. This comes with a canonical homomorphism γ: S → G by mapping (p, V) ↦ p^{−1} , and with a canonical action on X by mapping (p, V) ↦ α_{p}|_{V}. Let S be the quotient of S_{0} by the kernel of this map. That is, we consider two elements of S_{0} equivalent if they give the same element of G and the same partial homeomorphism on X. Now we take the groupoid of germs of the action of S on X with the germ relation from [28], that is, two elements s, t ∈ S have the same germ at x ∈ X if there is an idempotent e in S defined at x so that se = te.

Lemma 5.3. Assume that P is an Ore monoid and the subsets U_{p} are clopen. Then the groupoid X ⋊S above is canonically isomorphic to the groupoid model in Definition 4.12.

Proof. We map the free inverse semigroup S_{0} above to the inverse semigroup of bisections of the groupoid H in Definition 4.12 by mapping (p, V) to H_{1,p}^{1} ∩ s^{−1} (V); this is easily seen to be a bisection of H that acts on X = H^{0} by the partial homeomorphism α_{p}|_{V} and has degree p^{−1} . Thus the action of S_{0} on X and the homomorphism γ both factor through the inverse semigroup of bisections of H, where γ maps bisections contained in H_{g}^{1} to g and where the action of H on X is used to let bisections act on X. By construction, s ∈ S_{0} is annihilated by γ if and only if the corresponding bisection in H is contained in H_{1}. This groupoid comes from an equivalence relation, so a bisection is trivial if and only if it acts trivially on X. Hence s ∈ S_{0} becomes idempotent in S if and only if it is mapped to an idempotent bisection of H. This shows that the quotient S of S_{0} is exactly the image of S_{0} in the inverse semigroup of bisections of H. The groupoid H is covered by bisections that belong to S and are of the form

(H_{1,p}^{1} ∩ s^{−1} (V)) ∘ (H_{1,q}^{1} ∩ s^{−1} (W))^{−1}

for p, q ∈ P and V ⊆ U_{p}, W ⊆ U_{q} such that α_{p}|_{V} and α_{q}|_{W} are injective. It is not clear whether these bisections already form an inverse semigroup; but at least, since they cover the arrow space of H, any product of such bisections is again covered by bisections of H of this special form. Therefore, the inverse semigroup S and the inverse semigroup of bisections of H have the same germ groupoids attached to their actions on X, that is, H ≅ X ⋊ S. □

The examples considered by Exel and Renault in [31] are actions of (ℕ^{k}, +) by (globally defined) local homeomorphisms, so they certainly fit into our framework. This is remarkable because [31] also contains counterexamples where Exel’s interaction group approach to non-invertible dynamical systems does not apply. This leads Exel and Renault to speculate that something should go wrong in these counterexamples. Exel defines interactions in [27] as a way to describe dynamical systems that are non-deterministic in both past and future time directions. A local homeomorphism with a transfer operator is a particular example of an interaction, and the dynamics generated by a single local homeomorphism may be studied quite well using interactions. Exel proposed the concept of an interaction group in [29] in order to extend this to more general dynamical systems. In [31], Exel and Renault give rather simple examples of commuting local homeomorphisms S, T: X → X that cannot be embedded in an interaction group over ℤ^{2}. We are going to discuss this, assuming both S, T to be surjective because this happens in the counterexamples in [31]. The problem is the following. The local homeomorphism S generates an equivalence relation on X by x ∼_{S} y if S(x) = S(y), and similarly for T. If there is an interaction group, then these relations ∼_{S} and ∼_{T} must commute, that is, there is z ∈ X with x ∼_{S} z ∼_{T} y if and only if there is w ∈ X with x ∼_{T} w ∼_{S} y (see [31, Proposition 14.1]). There are, however, commuting endomorphisms S, T for which the relations ∼_{S} and ∼_{T} do not commute. So such S, T cannot be part of an interaction group. Why is this no problem for our groupoid model? Since our topological correspondences are already local homeomorphisms, our groupoid H has object space H^{0} = X. The group completion of (ℕ^{2}, +) is (ℤ^{2}, +), so the groupoid H is ℤ^{2}-graded, H = ⨆ _{g∈ℤ^{2}} H_{g}. A point in H_{g} is given by (x, y) ∈ X and n_{1}, n_{2}, m_{1}, m_{2} ∈ ℕ with S^{n_{1}} T^{n_{2}} (x) = S^{m_{1}} T^{m_{2}} (y) and (m_{1}, m_{2}) − (n_{1}, n_{2}) = g, and this is an arrow x ← y. Thus the range and source maps identify H_{g} with the union of the subsets

H_{n_{1} ,n_{2} ,m_{1} ,m_{2}} = {(x, y) ∈ X × X | S^{n_{1}} T^{n_{2}} (x) = S^{m_{1}} T^{m_{2}} (y)}.

We treat these as relations on X. The subsets H_{n_{1} ,n_{2} ,m_{1} ,m_{2}} are closed in X × X. If k_{1}, k_{2} ∈ ℕ then H_{n_{1} ,n_{2} ,m_{1} ,m_{2}} is both open and closed in H_{n_{1} +k_{1} ,n_{2} +k_{2} ,m_{1} +k_{1} ,m_{2} +k_{2}} because the map S^{k_{1}} T^{k_{2}} is a local homeomorphism, hence locally injective. The relation H_{n_{1} ,n_{2} ,m_{1} ,m_{2}} may also be interpreted as the graph of the multi-valued map S^{−n_{1}} T^{−n_{2}} S^{m_{1}} T^{m_{2}} on X. What happens when we compose our relations? We have (x, y) ∈ H_{k_{1} ,k_{2} ,l_{1} ,l_{2}} ∘ H_{l_{1} ,l_{2} ,m_{1} ,m_{2}} if and only if there is z ∈ X with S^{k_{1}} T^{k_{2}} (x) = S^{l_{1}} T^{l_{2}} (z) = S^{m_{1}} T^{m_{2}} (y). Since S^{l_{1}} T^{l_{2}} is surjective by assumption, the point z can always be found if S^{k_{1}} T^{k_{2}} (x) = S^{m_{1}} T^{m_{2}} (y), so

(5.4)  H_{k_{1} ,k_{2} ,l_{1} ,l_{2}} ∘ H_{l_{1} ,l_{2} ,m_{1} ,m_{2}} = H_{k_{1} ,k_{2} ,m_{1} ,m_{2}} .

Exel and Renault say that S and T star-commute if for all x, y ∈ X with T (x) = S(y) there is a unique z ∈ X with S(z) = x and T (z) = y. Under this assumption, they construct an interaction group containing S and T. In our notation, S and T star-commute if and only if H_{0,1,1,0} = H_{0,0,1,0} ∘ H_{0,1,0,0} ; the inclusion H_{0,1,1,0} ⊇ H_{0,0,1,0} ∘ H_{0,1,0,0} is trivial. The relation H_{0,1,1,0} describes the multi-valued map T^{−1} ∘ S, whereas the relation H_{0,0,1,0} ∘ H_{0,1,0,0} describes the multi-valued map S∘T^{−1} . So S and T star-commute if and only if T^{−1} ∘S = S ∘ T^{−1} . If this fails, there is no good way to define a topological correspondence or an interaction for the element (1, −1) ∈ ℤ^{2}. The difference between these two relations is, however, always small in the sense that

T ∘ (S ∘ T^{−1} ) = S ∘ (T ∘ T^{−1} ) = S = T ∘ (T^{−1} ∘ S).

Since x, y ∈ X with T (x) = T (y) are equivalent in our groupoid H, the difference between the relations S ∘ T^{−1} and T^{−1} ∘ S does not matter once we take the whole groupoid into account. Let us also examine this issue from the point of view of the Cuntz–Pimsner algebra of the resulting product system (E_{p})_{p∈ℕ^{2}} . The subspace H_{n_{1} ,n_{2} ,m_{1} ,m_{2}} of the groupoid corresponds to the subspace 𝕂(E_{(m_{1} ,m_{2} )} , E_{(n_{1} ,n_{2} )} ) of the Cuntz– Pimsner algebra, compare the proof of Theorem 4.17. Since we assume the maps S, T to be surjective, the correspondences E_{p} are full for all p ∈ ℕ^{2}. Hence

𝕂(E_{(m_{1} ,m_{2} )} , E_{(n_{1} ,n_{2} )} ) · 𝕂(E_{(k_{1} ,k_{2} )} , E_{(m_{1} ,m_{2} )} ) = 𝕂(E_{(k_{1} ,k_{2} )} , E_{(n_{1} ,n_{2} )} )

for all k_{1}, k_{2}, m_{1}, m_{2}, n_{1}, n_{2} ∈ ℕ. This corresponds to (5.4). The zero fibre of the Cuntz–Pimsner algebra is the inductive limit of the C^{∗}-subalgebras 𝕂(E_{n_{1} ,n_{2}} ) for n_{1}, n_{2} → ∞. In particular, 𝕂(E_{0,1}) and 𝕂(E_{1,0}) are contained in 𝕂(E_{1,1}). Although E_{1,1} = E_{1,0} ⊗_{C_{0} (X)} E_{0,1} ≅ E_{0,1} ⊗_{C_{0} (X)} E_{1,0}, we cannot expect in general that 𝕂(E_{0,1}) · 𝕂(E_{1,0}) is equal to 𝕂(E_{1,1}): this goes wrong if S and T do not star-commute. It may also happen that 𝕂(E_{0,1}) · 𝕂(E_{1,0}) ≠ 𝕂(E_{0,1}) · 𝕂(E_{1,0}) or, equivalently, that 𝕂(E_{0,1}) · 𝕂(E_{1,0}) is not a C^{∗}-algebra. Summing up, we have seen that the commutative semigroup ℕ^{2} may well generate some noncommutative phenomena both on the groupoid and C^{∗}-algebra level. So the reason why Cuntz–Pimsner algebras for proper product systems over ℕ^{2} are tractable is not that ℕ^{2} is commutative—it is that ℕ^{2} satisfies Ore conditions. To make this clearer, consider now an arbitrary semigroup P. The Cuntz– Pimsner algebra of a proper product system (E_{p})_{p∈P} over P must contain 𝕂(E_{p}) for all p ∈ P. Given p, q ∈ P, we therefore need a C^{∗}-algebra containing both 𝕂(E_{p}) and 𝕂(E_{q}). If P is an Ore monoid, then there is t ∈ P with t ≥ p, q, and then 𝕂(E_{p}) and 𝕂(E_{q}) are both contained in 𝕂(E_{t}). It is irrelevant for the construction how much of 𝕂(E_{t}) is generated by 𝕂(E_{p}) and 𝕂(E_{q}) or whether 𝕂(E_{p})·𝕂(E_{q}) is a C^{∗}-algebra. Indeed, we would not even ask for any relationship unless t were chosen minimal, t = p ∨ q, which only exists in lattice-ordered semigroups. The examples in [31] show that for commutative P the interaction group approach of Exel is trying implicitly to combine 𝕂(E_{p}) and 𝕂(E_{q}) in such a way that 𝕂(E_{p}) · 𝕂(E_{q}) = 𝕂(E_{q}) · 𝕂(E_{p}) is again a C^{∗}-algebra. This led Exel in [30] to study when A · B = B · A for two C^{∗}-subalgebras A and B of another C^{∗}-algebra. Finally, there is one thing where an interaction group helps. If we only have an action of ℕ^{2}, then we can only restrict it to submonoids of ℕ^{2}. An interaction group on ℤ^{2} may also be restricted to subgroups of ℤ^{2}; in the above notation, an interaction group gives well-defined topological correspondences S^{n_{1}} T^{n_{2}} for all n_{1}, n_{2} ∈ ℤ. Without it, we only have this if n_{1} and n_{2} have the same sign. Examples of such restrictions are the polymorphisms in [17]. These may, however, be written directly as topological graphs.

5.2. Semigroup C*-algebras. How to define the C^{∗}-algebra of a monoid P? A satisfactory answer is given by Xin Li [55], assuming P to be left cancellative. If P is Ore, we shall describe Xin Li’s C^{∗}-algebra as the Cuntz–Pimsner algebra of a product system. More precisely, we change the order of multiplication in P so as to get product systems over P instead of over P^{op}, compare Remark 3.1. Thus for us P is a right cancellative, right Ore monoid, and we describe the semigroup C^{∗}-algebra of P^{op} in the notation of [55]. The discussion below is closely related to the description of semigroup C^{∗}-algebras in [16]. Why is it non-trivial to construct semigroup C^{∗}-algebras? There is an obvious product system over any monoid: just take the complex numbers everywhere, with the obvious multiplication maps. A nondegenerate representation of this product system, however, is a representation of P by unitaries, not isometries. Hence the Cuntz–Pimsner algebra of this product system is the group C^{∗}-algebra of the group completion of P. To get an interesting Cuntz–Pimsner algebra, we need a non-trivial product system. So why not take the universal C^{∗}-algebra for representations of P by isometries? This is generated by one isometry s_{p} for each p ∈ P, with the relations s_{p}^{∗}s_{p} = 1 for all p ∈ P and s_{p}s_{q} = s_{qp} for all p, q ∈ P. This universal semigroup C^{∗}-algebra of P^{op} is introduced by Murphy [62]. It is, however, usually too “wild” to say much about it. It is rarely simple or exact. The right way to “tame” Murphy’s universal C^{∗}-algebra of a semigroup is to impose relations on the range projections of the isometries s_{p}. Xin Li [55] proposes a set of such relations modelled on properties of the regular representation of the semigroup. We are about to construct a product system (E_{p})_{p∈P} that gives Xin Li’s C^{∗}-algebra, and such that s_{p}^{∗} ∈ E_{p}. Thus s_{p}s_{p}^{∗} ∈ E_{1}. In our approach, the desired relations among the range projections s_{p}s_{p}^{∗} are encoded in the C^{∗}-algebra D := E_{1}. Let D be the C^{∗}-subalgebra of ℓ^{∞} (P) generated by the characteristic functions of left ideals of the form P pq^{−1} ∩ P for p, q ∈ P, where

P pq^{−1} ∩ P = {x ∈ P | ∃y ∈ P: xq = yp}.

In the following, we will use the group completion G of P and write P g for g ∈ G. Since P ⊆ G is right cancellative, we have xq ∈ P p if and only if xqt ∈ P pt for some t ∈ P; thus P g for g ∈ G is well-defined, that is, does not depend on how we write g = pq^{−1} for p, q ∈ P. Since D is commutative and unital, it is of the form C(X) for a compact space X. Right translation by p ∈ P maps P g ∩ P to P gp ∩ P p. The characteristic function of this intersection is the product of the characteristic functions of P gp ∩ P and P p ∩ P and hence belongs to D. Thus right translation by p gives an endomorphism φ_{p}: D → D. These maps form an action of P^{op} on D by endomorphisms, that is, φ_{p} ∘ φ_{q} = φ_{qp} for all p, q ∈ P. The endomorphism φ_{p} for p ∈ P maps the constant function 1 to the characteristic function e_{p} of the subset P · p ⊆ P. Actually, φ_{p}: D → e_{p}D is an isomorphism because the map

P → P · p, x ↦ x · p,

is bijective and this map and its inverse map the generators of D to the characteristic functions of subsets of the form P g ∩ P p, which are exactly all the generators of e_{p}D. We may turn the action of P^{op} by the endomorphisms (φ_{p})_{p∈P} into a product system (E_{p})_{p∈P} over P as in Remark 3.1, with reversed order of multiplication in P. Explicitly, E_{p} = φ_{p}(D) · D = φ_{p}(1_{D}) · D = e_{p}D as a Hilbert D-module, with left action of D through φ_{p}, and the multiplication maps are E_{p} ⊗_{D} E_{q} ⟶^{∼} E_{pq}, x⊗y ↦ φ_{q}(x)·y. The Cuntz–Pimsner C^{∗}-algebra of this product system is canonically isomorphic to the semigroup crossed product for the action (φ_{p})_{p∈P} of P^{op} on D (see [33, Section 3]), and this is isomorphic to Xin Li’s semigroup C^{∗}-algebra of P^{op} by [55, Lemma 2.14]. Explicitly, the isomorphism looks as follows. The Cuntz–Pimsner algebra O of (E_{p})_{p∈P} is generated by D and copies S_{p}(E_{p}) of E_{p} for each p ∈ P. Since E_{p} = e_{p} · D and S_{p}(e_{p}d) = S_{p}(e_{p})d for all d ∈ D, O is already generated by D and the elements s_{p} = S_{p}(e_{p})^{∗}. Since ⟨e_{p}, e_{p}⟩_{E_{p}} = e_{p} and |e_{p}⟩⟨e_{p}| = id_{E_{p}}, the element s_{p} is an isometry with range projection e_{p}. Furthermore, s_{p}^{∗} · s_{q}^{∗} = s_{pq}^{∗} or, equivalently, s_{p} · s_{q} = s_{qp} for all p, q ∈ P. As it turns out, the relations of D and the relations s_{p}s_{p}^{∗} = e_{p}, s_{p}^{∗}s_{p} = 1, s_{p} · s_{q} = s_{qp} for p, q ∈ P imply all relations for the Cuntz–Pimsner algebra of (E_{p})_{p∈P} . Thus the Cuntz–Pimsner algebra agrees with Xin Li’s semigroup C^{∗}-algebra of P^{op}.

Now we describe a groupoid model for our Cuntz–Pimsner algebra. Such groupoid models are already constructed in [56], even under weaker assumptions on the semigroup P. The projection e_{p} in C(X) corresponds to a clopen subset V_{p} ⊆ X. The isomorphism φ_{p}: D → e_{p}D corresponds to a homeomorphism r_{p}: V_{p} → X. More precisely, φ_{p}(x)|_{V_{p}} = x ∘ r_{p} and φ_{p}(x)|_{X\V_{p}} = 0 for all x ∈ D. Let s_{p}: V_{p} → X be the inclusion map; this is a homeomorphism onto a clopen subset and hence a local homeomorphism. Thus (V_{p}, r_{p}, s_{p}) is a proper topological correspondence on X. The resulting correspondence on D = C(X) is C(V_{p}) = e_{p}D with the obvious right Hilbert D-module structure and the left D-action φ_{p} = r_{p}^{∗}. This is equal to the C^{∗}-correspondence E_{p} associated to the endomorphism φ_{p}. Now identify V_{p} ≅ X through r_{p} and rewrite our topological correspondence in the form (X, id_{X} , θ_{p}), where θ_{p}: X → U_{p} ⊆ X applies the inverse r_{p}^{−1} ; these topological correspondences are as in (5.1), where U_{p} = X and θ_{p} is a homeomorphism onto a clopen subset of X. We have θ_{p} ∘ θ_{q} = θ_{qp} for all p, q ∈ P because φ_{p} ∘ φ_{q} = φ_{qp}. Hence we get an action of P^{op} on X by partial homeomorphisms, which is an action of the type considered in Section 5.1. The resulting product system over P is canonically isomorphic to the product system (E_{p})_{p∈P} associated to the action (φ_{p})_{p∈P} of P^{op} on D = C(X) by ∗-endomorphisms. As in Section 5.1, we get a groupoid model for the Cuntz–Pimsner algebra of the product system (E_{p})_{p∈P} , and this groupoid model is the groupoid of germs for the pseudogroup of partial homeomorphisms of X generated by the partial homeomorphisms θ_{p} for p ∈ P. Finally, we mention a quicker alternative definition of Xin Li’s semigroup C^{∗}-algebras, see also [56] for an extension of this approach to more general semigroups. The main result of [50] shows that any semigroup crossed product is a full corner in the crossed product for a group action on a larger C^{∗}-algebra. In our case, this larger C^{∗}-algebra is the C^{∗}-subalgebra A of ℓ^{∞} (G) generated by the characteristic functions of subsets of the form P g ⊆ G for g ∈ G. The right translation action of G on ℓ^{∞} (G) restricts to an action of G on A by automorphisms. This also induces an action of G by homeomorphisms on the spectrum Y of the C^{∗}-algebra A. The C^{∗}-algebra D is the full corner in A corresponding to the projection 1_{P}, the characteristic function of P ⊆ G. The action (φ_{p}) on D above is the compression of the action of G on A. Hence the semigroup crossed product discussed above is canonically isomorphic to the full corner 1_{P}(A ⋊ G)1_{P} in the crossed product A ⋊ G. Similarly, the groupoid model for the action of P on X is the restriction of the transformation groupoid Y ⋊ G to the compact-open subset X ⊆ Y.

6. Properties of the groupoid model

Let an Ore monoid P act on a locally compact space X by topological correspondences (M_{p}, σ_{p,q}). The Cuntz–Pimsner algebra of the resulting product system over P is identified with a groupoid C^{∗}-algebra C^{∗}(H) in Theorem 4.17.

Many properties of C^{∗}(H) are equivalent or closely related to properties of the underlying groupoid H. We harvest some known results of this type regarding nuclearity, simplicity or ideal structure, tracial and KMS weights, and pure infiniteness. One interesting aspect of our groupoid model is that it involves a “precompiler”: given an action of an Ore monoid P by topological correspondences on a space X, we first construct an action of P on another space H^{0} by local homeomorphisms, and then we take the Cuntz–Pimsner algebra of this new action. Hence any C^{∗}-algebra that may be obtained as the Cuntz–Pimsner algebra of some action of P by topological correspondences may also be obtained from an action of P on another space by local homeomorphisms. Therefore, for some purposes we may assume without loss of generality that we are dealing with an action of P by local homeomorphism. In this section, however, the main point is to rewrite properties of the groupoid H in terms of the original action on X. For actions of P by local homeomorphisms, what we are going to do is already wellknown. Our criteria simplify further if the space X is discrete; this happens, in particular, for higher-rank graphs. We begin with quick criteria for separability, unitality and nuclearity.

Remark 6.1. The groupoid C^{∗}-algebra of an étale locally compact groupoid is separable if and only if the underlying groupoid is second countable. This happens if and only if the closed subspace X′ ⊆ X is second countable and the group G is countable. This follows if X is second countable and P is countable; in the latter case, we can see directly that the Cuntz–Pimsner algebra is separable.

Remark 6.2. The C^{∗}-algebra C^{∗}(H) is unital if and only if H^{0} is compact. This happens if and only if the closed subspace X′ ⊆ X is compact because the projection map π_{1} : H^{0} → X is a continuous, proper map with image X′ (see Lemma 4.9 and the discussion after Definition 4.15).

Theorem 6.3. The full groupoid C^{∗}-algebra C^{∗}(H) is nuclear if and only if the groupoid H is topologically amenable. In that case, C^{∗}(H) belongs to the bootstrap class. The groupoid H_{1} ⊆ H is always topologically amenable. If G is amenable, then the groupoid H is also amenable, and C^{∗}(H_{1}) belongs to the bootstrap class.

Proof. An étale, Hausdorff, locally compact groupoid is (topologically) amenable if and only if its reduced C^{∗}-algebra is nuclear by [4, Corollary 6.2.14]. Furthermore, if H is amenable, then its reduced and full C^{∗}-algebras coincide, so the full one is also nuclear. Conversely, if the full groupoid C^{∗}-algebra is nuclear, then so is the reduced one because nuclearity is hereditary for quotients. Hence nuclearity of the full groupoid C^{∗}-algebra is also equivalent to amenability of the groupoid. Any amenable groupoid is “a-T-menable” by [76, Lemma 3.5]; that is, it acts properly and isometrically on a continuous field of affine Euclidean spaces. The proof of the Baum–Connes conjecture for a-T-menable groupoids also shows that their groupoid C^{∗}-algebras belong to the bootstrap class, see [76, Proposition 10.7]. Since C_{0}(X) is nuclear, Theorem 3.21 shows that the unit fibre O_{1} in the associated Cuntz–Pimsner algebra is always nuclear; then O itself is nuclear if G is amenable. Theorem 4.17 and its proof identify O_{1} and O with C^{∗}(H_{1}) and C^{∗}(H), respectively. So the statements about amenability of H_{1} and H follow from the first sentence in the theorem. It is elementary to prove the topological amenability of H_{1} directly. The open subgroupoids H_{p,p}^{1} for p ∈ P are proper equivalence relations. So we may normalise the counting measure on the fibres of H_{p,p}^{1} to give an invariant mean on H_{p,p}^{1}. When we view these invariant means on H_{p,p}^{1} as means on H_{1} for p in the filtered category C_{P}, we get an approximately invariant mean on H_{1}. □

We have not yet tried to characterise amenability of H in terms of the original action by topological correspondences.

6.1. Open invariant subsets and minimality.

Definition 6.4. A topological groupoid H is minimal if H^{0} has no open, invariant subsets besides ∅ and H^{0}.

Being minimal is a necessary condition for C^{∗}(H) to be simple because open invariant subsets of H^{0} generate ideals in C^{∗}(H). We are going to describe the open, invariant subsets of H^{0} in terms of the original data (M_{p}, σ_{p,q}). The following lemma gives a base for the topology on H^{0} and will also be used for other purposes.

Lemma 6.5. For p ∈ P and an open subset U ⊆ M_{p}, let

π_{p}^{−1} (U) := {(m_{q})_{q∈P} ∈ H^{0} | m_{p} ∈ U} ⊆ H^{0}.

The family B of subsets of this form is a base for the topology on H^{0} and, for each x ∈ H^{0}, the subsets π_{p}^{−1} (U) with x ∈ U form a neighbourhood base at x. This base for the topology is closed under finite unions, finite intersections, and under applying s̃_{t}^{−1} for all t ∈ P; and

(6.6)  s̃_{t}(π_{t}^{−1} (U)) = π_{1}^{−1} (s_{t}(U))  for all t ∈ P and U ⊆ M_{t} open.

Proof. By definition of the product topology, intersections ⋂ _{p∈F} π_{p}^{−1} (U_{p}) for finite subsets F ⊆ P and open subsets U_{p} ⊆ M_{p} for p ∈ F form a base of the topology on H^{0}, and such intersections with x ∈ π_{p}^{−1} (U_{p}) form a neighbourhood base for x ∈ H^{0}. If B is closed under finite intersections, then B itself is this canonical base of the topology, and similarly for neighbourhoods of x. Let p, q ∈ P. Then r_{p,q}(m_{pq}) = m_{p} for all (m_{t})_{t∈P} ∈ H^{0}. Thus

(6.7)  π_{pq}^{−1} (r_{p,q}^{−1}(U)) = π_{p}^{−1} (U)

for each open subset U ⊆ M_{p}. Since the maps r_{p,q} are continuous, r_{p,q}^{−1}(U) is again open. Now we consider a finite intersection ⋂ _{i=1}^{n} π_{p_{i}}^{−1} (U_{i}) for F = {p_{1}, …, p_{n}} ⊆ P and U_{i} = U_{p_{i}} ⊆ M_{p_{i}} . Since P is a right Ore monoid, there are p ∈ P and q_{i} ∈ P with p_{i}q_{i} = p for i = 1, …, n. Then π_{p_{i}}^{−1} (U_{i}) = π_{p}^{−1} (r_{p_{i} ,q_{i}}^{−1} (U_{i})). Thus

⋂_{i=1}^{n} π_{p_{i}}^{−1} (U_{i}) = π_{p}^{−1} ( ⋂_{i=1}^{n} r_{p_{i} ,q_{i}}^{−1} (U_{i})) ,

⋃_{i=1}^{n} π_{p_{i}}^{−1} (U_{i}) = π_{p}^{−1} ( ⋃_{i=1}^{n} r_{p_{i} ,q_{i}}^{−1} (U_{i})) .

Thus B is closed under finite intersections and finite unions. We have

(6.8)  s̃_{t}^{−1} (π_{p}^{−1} (U)) = π_{tp}^{−1} (s_{t,p}^{−1} (U))

because s̃_{t}((m_{p})) = (s_{t,p}(m_{tp}))_{p∈P} . Thus B is closed under s̃_{t}^{−1} for all t ∈ P. Lemma 4.10 shows that (π_{t}, s̃_{t}): H^{0} → M_{t} ×_{s_{t},X,π_{1}} H^{0} is a homeomorphism. Given U ⊆ M_{t}, consider the set of all ω ∈ H^{0} for which there is m ∈ U with (m, ω) ∈ M_{t} ×_{s_{t},X,π_{1}} H^{0}. By definition of the fibre product, this is π_{1}^{−1} (s_{t}(U)). The homeomorphism (π_{t}, s̃_{t}) shows, however, that it is also s̃_{t}(π_{t}^{−1} (U)). This gives (6.6). □

Definition 6.9. The indicator of an invariant subset A ⊆ H^{0} is the following subset of X:

A^{#} := {x ∈ X | π_{1}^{−1} (x) ⊆ A}.

Let X′ ⊆ X be the subset of possible situations. By definition, any saturated subset contains X \ X′. So we lose no information if we restrict attention to A^{#} ∩ X′.

Definition and Lemma 6.10. The indicator B of an open invariant subset is open in X and has the following two properties: hereditary: if p ∈ P, m ∈ M_{p} satisfy r_{p}(m) ∈ B, then s_{p}(m) ∈ B; saturated: if p ∈ P, x ∈ X satisfy s_{p}(m) ∈ B for all m ∈ r_{p}^{−1} (x), then

x ∈ B. If S ⊆ P is a subset that generates P, then a subset B ⊆ X is hereditary and saturated if and only if it satisfies the above two conditions for all p ∈ S.

Proof. Let B = A^{#} . First we show that A^{#} is open. By definition, X \ A^{#} = π_{1}(H^{0} \ A). The map π_{1} : H^{0} → X is proper by Lemma 4.9. So it maps the closed subset H^{0} \ A to a closed subset of X. Hence X \ A^{#} is closed and A^{#} is open. Next we check that A^{#} is hereditary. If s_{p}(m) ∉ A^{#} , then there is η ∈ H^{0} \ A with π_{1}(η) = s_{p}(m). The concatenation m · η ∈ H^{0} exists because s_{p}(m) = π_{1}(η), and has π_{1}(m · η) = r_{p}(m). Since A is invariant, so is H^{0} \ A. Hence m · η ∉ A and r_{p}(m) ∉ A^{#} . We check that A^{#} is saturated. Assume s_{p}(m) ∈ A^{#} for all m ∈ r_{p}^{−1} (x), and let η ∈ H^{0} satisfy π_{1}(η) = x. Decompose η as η = m · η′ with η′ ∈ H^{0} and m := π_{p}(η) by Lemma 4.10. Since r_{p}(m) = π_{1}(η) = x, the assumption gives π_{1}(η′) = s_{p}(m) ∈ A^{#} , so η′ ∈ A. Since A is invariant, we get η = m · η′ ∈ A and x ∈ A^{#} .

If a subset B ⊆ X satisfies the conditions of being hereditary and saturated for given p, q ∈ P, then it also satisfies them for p·q because M_{pq} ≅ M_{p}×_{s_{p} ,X,r_{q}} M_{q}. Hence it suffices to verify these conditions for a set of generators for P. □

Theorem 6.11. The complete lattice of open H-invariant subsets of H^{0} is isomorphic to the complete lattice of open, hereditary, saturated subsets of X. In one direction, the isomorphism maps an open H-invariant subset A ⊆ H^{0} to its indicator; in the other direction, it maps an open, hereditary, saturated subset B ⊆ X to

A := ⋃_{p∈P} (s_{p} ∘ π_{p})^{−1} (B) = {(m_{p})_{p∈P} ∈ H^{0} | ∃p: s_{p}(m_{p}) ∈ B}.

Proof. Lemma 6.10 shows that the indicator A^{#} of an open invariant subset A ⊆ H^{0} is open, hereditary and saturated. Conversely, let B ⊆ X be open, hereditary and saturated, and define A ⊆ H^{0} as above. This is clearly open. We first check that A is invariant; then we check that its indicator is B. Let η ∈ H^{0}. Then s_{p}∘π_{p}(η) ∈ X is the situation at time p ∈ P in the complete history η. Thus A consists of all complete histories that, at some time, visit B ⊆ X. However, if s_{p}∘π_{p}(η) ∈ B, then s_{pq}π_{pq}(η) ∈ B for all q ∈ P because B is hereditary; indeed, π_{pq}(η) ∈ M_{pq} ≅ M_{p} ×_{s_{p} ,X,r_{q}} M_{q} corresponds to a pair (π_{p}(η), m_{q}) with r_{q}(m_{q}) = s_{p}π_{p}(η) ∈ B, so s_{pq}(π_{pq}(η)) = s_{q}(m_{q}) ∈ B. The subset pP ⊆ P is cofinal. Hence η ∈ A if and only if the set of q ∈ P with s_{q} ∘ π_{q}(η) ∈ B is cofinal in P. A subset A of H^{0} is H-invariant if and only if s̃_{t}^{−1} (A) = A for all t ∈ P. Let η ∈ H^{0} and write it as η = m · η′ for m ∈ M_{t}, η′ ∈ H^{0} with s_{t}(m) = r_{t}(η′). Thus s̃_{t}(η) = η′. Then s_{tp}π_{tp}(η) = s_{p}π_{p}(η′) for all p ∈ P. So if there is p ∈ P with s_{p}π_{p}(η′) ∈ B, then there is q ∈ P with s_{q}π_{q}(η) ∈ B, namely, q = tp; conversely, if there is q ∈ P with s_{q}π_{q}(η) ∈ B, then the set of such q ∈ P is cofinal and hence contains some element of the form tp with p ∈ P by (O1). Then s_{p}π_{p}(η′) ∈ B. This shows that η ∈ A if and only if η′ ∈ A. So A is invariant as desired. Now we check that the indicator of A is B. By construction, if x ∈ B, then π_{1}^{−1} (x) ⊆ A. Conversely, let x ∈ X \ B. We must construct η ∈ H^{0} \ A with π_{1}(η) = x. For each p ∈ P, there is m_{p} ∈ M_{p} with r_{p}(m_{p}) = x and s_{p}(m_{p}) ∉ B because otherwise B would not be saturated. Since X \ X′ ⊆ B, and s_{p}(m_{p}) ∉ B, there is η_{p} ∈ H^{0} with π_{p}(η_{p}) = m_{p} and hence π_{1}(η_{p}) = x and s_{p} ∘ π_{p}(η_{p}) ∉ B. Since B is open and the map π_{p} is proper by Lemma 4.9, the set K_{p} of all such η_{p} is a compact subset of H^{0}. We have seen above that s_{p}π_{p}(η) ∈ B implies s_{pq}π_{pq}(η) ∈ B for all p, q ∈ P. Hence K_{p} ⊇ K_{pq} for all p, q ∈ P. Since P is Ore, this tells us that {K_{p}}_{p∈P} is a directed set of compact, nonempty subsets in H^{0}. The intersection of such a family of subsets is non-empty. A point η in the intersection satisfies π_{1}(η) = x and s_{p}π_{p}(η) ∉ B for all p ∈ P, so that η ∉ A. Thus A^{#} = B.

We have constructed two maps A ↦ B and B ↦ A from open invariant subsets of H^{0} to open, hereditary, saturated subsets of X and back, and we have seen that the composite map B ↦ A ↦ B is the identity, that is, the indicator of the subset A defined in the theorem is the given subset B. Conversely, let A′ ⊆ H^{0} be an invariant open subset. Let B be its indicator and define A ⊆ B as in the statement of the theorem. We must show that A′ = A. Since B is the indicator of A′, we have π_{1}^{−1} (B) ⊆ A′. Since A′ is invariant, this implies (s_{p}π_{p})^{−1} (B) ⊆ A′ for all p ∈ P, that is, A ⊆ A′. It remains to prove A′ ⊆ A. So we take η ∈ A′. Since A′ is open, Lemma 6.5 gives p ∈ P and an open subset U ⊆ M_{p} such that π_{p}^{−1} (U) ⊆ A′. If η′ ∈ H^{0} satisfies π_{1}(η′) ∈ s_{p}(U), then there is m ∈ U ⊆ M_{p} with s_{p}(m) = π_{1}(η′), so m · η′ ∈ H^{0} is well-defined; it belongs to π_{p}^{−1} (U) ⊆ A′ by construction. Since s̃_{p}(m · η′) = η′ and A′ is invariant, we get η′ ∈ A′ for all η′ ∈ H^{0} with π_{1}(η′) ∈ s_{p}(U). Thus s_{p}(U) is contained in the indicator B of A′. Then π_{p}^{−1} (U) ⊆ A, so η ∈ A as desired. □

Corollary 6.12. The groupoid H is minimal if and only if X has no nontrivial open, hereditary, saturated subsets. □

Example 6.13. Let P = (ℕ, +) and let r_{1} be surjective, so that we are dealing with a regular topological graph (see the beginning of Section 5). Then our notion of a hereditary and saturated subset of the space of vertices X is the same one used by Katsura [45] to describe the gauge-invariant ideals of a topological graph C^{∗}-algebra in the regular case.

Example 6.14. Let P = (ℕ^{k}, +) and assume that X is discrete and r_{p}: M_{p} → X is surjective for all p ∈ ℕ^{k}. Our data is equivalent to that of a row-finite k-graph without sources, and our Cuntz–Pimsner algebra is the higher-rank graph C^{∗}-algebra in this case. Since the standard basis e_{1}, …, e_{k} generates ℕ^{k}, a subset is hereditary and saturated if and only if it satisfies the conditions in Definition 6.10 for p = e_{1}, …, e_{k}. Our notion of being hereditary and saturated is equivalent to the one used by Raeburn, Sims and Yeend in [68] to describe the gauge-invariant ideals in a higher-rank graph C^{∗}-algebra. Theorem 6.23 below will show that, in general, the open invariant subsets of H^{0} correspond to those ideals in C^{∗}(H) that are “gauge-invariant” in a suitable sense.

6.2. Effectivity.

Definition 6.15. An étale topological groupoid H is essentially free if the subset of objects with trivial isotropy is dense in H^{0}. It is effective if every open subset of H^{1} \ H^{0} contains an arrow x with s(x) ≠ r(x). Equivalently, the interior of the set {h ∈ H^{1} \ H^{0} | r(h) = s(h)} is empty.

For a second countable, locally compact, étale groupoid, being effective or essentially free are equivalent properties by [72, Proposition 3.6] or [9, Lemma 3.1] (these articles use the name “topologically principal” for “essentially free”). Being essentially free is a variant of the aperiodicity condition that is used to characterise when topological higher-rank graph C^{∗}-algebras are simple (see, for instance, [79, Definition 5.2]). We cannot, however check whether H is essentially free without looking at points in H^{0}, that is, infinite paths. As we shall see, the following definition characterises when the groupoid H is effective in terms of the original data of an action by topological correspondences:

Definition 6.16. An action (M_{p}, σ_{p,q}) of an Ore monoid P on a locally compact Hausdorff space X by proper topological correspondences is effective if for all p, q ∈ P with pq^{−1} ≠ 1 in G and for all non-empty, open subsets U ⊆ X′, there are a, f, g ∈ P with paf = qag and y ∈ M_{paf} = M_{qag} with r_{paf} (y) ∈ U and mid_{p,a,f} (y) ≠ mid_{q,a,g} (y) in M_{a}. Here X′ ⊆ X denotes the closed subset of possible situations; mid_{p,a,f} (y) denotes the component in the middle factor M_{a} after identifying M_{paf} ≅ M_{p} ×_{X} M_{a} ×_{X} M_{f} , and similarly for mid_{q,a,g} (y).

Similar criteria for boundary path groupoids of higher-rank topological graphs being effective have been found by Wright [77]; for higher-rank graphs without topology, such criteria are also given in [40, 53, 74]. Before we explain the relationship between our criterion and others, we prove the theorem suggested by our notation:

Theorem 6.17. The groupoid H is effective if and only if the action (M_{p}, σ_{p,q}) is effective.

Proof. We may assume without loss of generality that X = X′. First we assume that the action (M_{p}, σ_{p,q}) is not effective. This means that there are p, q ∈ P with pq^{−1} ≠ 1 in G and a non-empty open subset U ⊂ X′ such that mid_{p,a,f} (y) = mid_{q,a,g} (y) in M_{a} for all a, f, g ∈ P with paf = qag and all y ∈ M_{paf} with r_{paf} (y) ∈ U. This means that (s̃_{p}x)_{a} = (s̃_{q}x)_{a} for all x ∈ π_{1}^{−1} (U) and all a ∈ P. Hence s̃_{p}x = s̃_{q}x in H^{0} for all x ∈ π_{1}^{−1} (U). Thus the elements of the form (x, pq^{−1} , x) for x ∈ π_{1}^{−1} (U) form a bisection B in H^{1} \ H^{0} with r|_{B} = s|_{B}, which means that H is not effective. Now assume that the action (M_{p}, σ_{p,q}) is effective. Let U ⊆ H^{1} \ H^{0} be a non-empty open subset. We need to find (x, g, y) ∈ U with x ≠ y. The intersection U ∩ H_{p,q}^{1} is non-empty for some p, q ∈ P. Replacing U by U ∩ H_{p,q}^{1}, we may arrange that U ⊆ H_{p,q}^{1}. The subgroupoid H_{1} is an increasing union of equivalence relations, so it is certainly effective. Hence we are done if pq^{−1} = 1 in G. Thus we may assume from now on that pq^{−1} ≠ 1 in G. We may shrink U to a bisection because H is étale. We may then shrink further so that r(U) = π_{t}^{−1} (U_{t}) for some t ∈ P and some non-empty open subset U_{t} ⊆ M_{t} because subsets of the form π_{t}^{−1} (U_{t}) form a base for the topology on H^{0} by Lemma 6.5. Since the map s_{t}: M_{t} → X is a local homeomorphism, we may shrink U_{t} even further, so that s_{t} becomes injective on U_{t}. Hence s_{t} restricts to a homeomorphism from U_{t} ⊆ M_{t} onto an open subset s_{t}(U_{t}) ⊆ X. We are going to show that there is x ∈ π_{t}^{−1} (U_{t}) with s̃_{p}x ≠ s̃_{q}x. Thus (x, g, x) is not an arrow in H; since r(U) = π_{t}^{−1} (U_{t}) and U ⊆ H_{p,q}^{1}, there must be y ∈ H^{0} with (x, g, y) ∈ U ⊆ H_{p,q}^{1}. Since s̃_{p}x = s̃_{q}y ≠ s̃_{q}x, we have x ≠ y, as desired.

Since P is Ore, we may find h, i ∈ P with ph = ti. Then we may find h′, i′ ∈ P with qhh′ = ti′. Thus phh′ = tih′ and qhh′ = ti′. Since H_{p,q}^{1} ⊆ H_{phh′,qhh′}^{1} , we may replace (p, q) by (phh′, qhh′). Thus we may assume without loss of generality that there are p′, q′ ∈ P with p = tp′ and q = tq′. Recall that s_{t} restricts to a homeomorphism from U_{t} onto V := s_{t}(U_{t}). Hence x ↦ s̃_{t}x is a homeomorphism from π_{t}^{−1} (U_{t}) onto π_{1}^{−1} (V) (compare Lemma 4.10). By assumption, there are a, f, g ∈ P with p′af = q′ag and y ∈ M_{p′af} with r_{p′af} (y) ∈ V and mid_{p′,a,f} (y) ≠ mid_{q′,a,g} (y) in M_{a}. By construction, there is z ∈ M_{t} with s_{t}(z) = r_{p′af} (y). Then (z, y) ∈ M_{t} ×_{X} M_{p′af} ≅ M_{tp′af} . We have s̃_{p}(z, y) = s̃_{p′}(y) and s̃_{q}(z, y) = s̃_{q′}(y) because p = tp′ and q = tq′. The M_{a}-component of s̃_{p′}(y) is mid_{p′,a,f} (y), and that of s̃_{q′}(y) is mid_{q′,a,g} (y). Since these are different, s̃_{p}(x) ≠ s̃_{q}(x) for any x ∈ H^{0} with tp′afcomponent (z, y). Such x exist because we have restricted to possible histories throughout, making the maps π_{p}: H^{0} → M_{p} surjective for all p ∈ P. □

Theorem 6.18. Assume that P is countable and X is second countable or, more generally, that H is second countable. The Cuntz–Pimsner algebra O or, equivalently, the C^{∗}-algebra C^{∗}(H), is simple if and only if the following three conditions are satisfied:

(1) C^{∗}(H) = C_{r}^{∗}(H);

(2) the action (M_{p}, σ_{p,q}) is effective;

(3) any non-empty, closed, hereditary, saturated of X contains X′.

The first condition above follows if H is amenable and, in particular, if G is amenable.

Proof. We use [9, Theorem 5.1], which characterises when the groupoid C^{∗}-algebra of a second countable, locally compact, Hausdorff, étale groupoid is simple. The groupoid H is always locally compact, Hausdorff, and étale. The third condition is equivalent to the minimality of H by Corollary 6.12. Since H is second countable, it is essentially free if and only if it is effective by [9, Lemma 3.1]. This is equivalent to the effectivity of the action (M_{p}, σ_{p,q}) by Theorem 6.17. Thus our conditions are equivalent to the three conditions in [9, Theorem 5.1]. □

Kwaśniewski and Szymański [49] also provide an aperiodicity criterion and use it to prove simplicity results and a uniqueness theorem for certain Cuntz– Pimsner algebras over Ore monoids. On the one hand, their criterion still works if the unit fibre A of a product system is only a liminal C^{∗}-algebra. On the other hand, it does not imply the simplicity of the Cuntz algebra O_{n} because it only uses the (partial, multivalued) action of P on the spectrum of A induced by the product system. This contains no information if A = ℂ as in the standard construction of O_{n}. Now we simplify the definition of being effective for the monoids P = (ℕ^{k}, +). Some of the steps also work for other monoids.

Lemma 6.19. The definition of an effective action of ℕ^{k} does not change if we assume, in addition, that the pair (p, q) ∈ ℕ^{k} is reduced, that is, there is no t ∈ ℕ^{k} with p, q ∈ ℕ^{k}t.

Proof. Take p = p′t, q = q′t for p′, q′, t ∈ ℕ^{k}. Since U ⊆ X′ is non-empty and consists of possible situations, there is m ∈ M_{t} with r_{t}(m) ∈ U. Since r_{t} is continuous, there is an open neighbourhood V of m with r_{t}(V) ⊆ U, and then U′:= s_{t}(V) is open as well. Let a, f, g and y′ verify the effectivity criterion for p′, q′, U′. Then the same a, f, g and y = m · y′ verify it for p, q, U. □

The role of f, g in Definition 6.16 is merely as padding to make paf = qag. In a commutative monoid such as ℕ^{k}, we may simply take f = q and g = p; or we may take f = (p ∨ q) − p and g = (p ∨ q) − p, where p ∨ q denotes the maximum of p and q in ℕ^{k}. The latter choice is minimal and may therefore be optimal.

Lemma 6.20. Let P = ℕ^{k}. Then the action on X is effective if and only if, for all reduced p, q ∈ ℕ^{k} with p − q ≠ 0 in ℤ^{k} and for all non-empty, open subsets U ⊆ X′, there are a ∈ ℕ^{k} and y ∈ M_{p+a+f} = M_{q+a+g} with r_{p+a+f} (y) ∈ U and mid_{p,a,f} (y) ≠ mid_{q,a,g} (y) in M_{a}, with f = (p ∨ q) − q and g = (p ∨ q) − p.

Proof. The criterion in the lemma differs from Definition 6.16 in two ways. First, we assume (p, q) reduced, which makes no difference by Lemma 6.19. Secondly, we choose particular f, g. To see that this makes no difference, choose f′, g′, y′ as in Definition 6.16. Then f′− f = g′− g = h ∈ ℕ^{k}. The truncation y := r_{p+a+f} (y′) ∈ M_{p+a+f} still has the same mid-part in M_{a} and hence also verifies the criterion in Definition 6.16. □

The condition in Lemma 6.20 is exactly Condition (ii) in [77, Theorem 3.1]. As shown in [77], this condition is equivalent to Yeend’s aperiodicity condition (A), which characterises when the groupoid model of the topological higher-rank graph is essentially free; Wright also gives an example where her finite-path version of aperiodicity is much easier to check than the original criterion.

Proposition 6.21. Let P = (ℕ, +) and assume X = X′. Then the action of P on X is effective if and only if the set of base points of loops without entrances has empty interior in X.

We will explain during the proof what loops without entrances are. The condition we arrive at characterises when the groupoid model for a regular topological graph is effective, compare [45, Definition 6.6].

Proof. Any reduced pair in ℕ is of the form (p, 0) or (0, p). Since our notion is symmetric, we may as well take q = 0. Thus the condition in Lemma 6.20 says that for any non-empty, open subset U ⊆ X′ and any p ∈ ℕ there is a ∈ ℕ and y ∈ M_{pa} with r_{pa}(y) ∈ U and mid_{p,a,0} (y) ≠ mid_{0,a,p} (y) in M_{a}. If there is y ∈ M_{p} with r_{p}(y) ∈ U and s_{p}(y) ≠ r_{p}(y), then a = 0 and y will do because the relevant mid-parts are s_{p}(y) and r_{p}(y), respectively. Thus we may assume as well that s_{p}(m) = r_{p}(m) for all m ∈ M_{p} with r_{p}(m) ∈ U; such a path is called a loop, and r_{p}(m) ∈ X is its base point.

Take m ∈ M_{p} with r_{p}(m) ∈ U. Identify M_{p} ≅ M_{1}^{×_{X} p} and write m as a path

x_{1} ⟵^{m_{1}} x_{2} ⟵^{m_{2}} x_{3} ⟵^{m_{3}} x_{4} → ⋯⟵^{m_{p}} x_{p+1} = x_{1}

with x_{1}, …, x_{p} ∈ X, m_{1}, …, m_{p} ∈ M_{1}. An entrance for this loop is 1 ≤ i ≤ p and an arrow m′: x′ → x_{i} with m′ ≠ m_{i}. If t is such an entrance, then take a = i and let y ∈ M_{pa} be the concatenation of the loop m and the path

x_{1} ⟵^{m_{1}} x_{2} ⟵^{m_{2}} x_{3} ← ⋯← x_{i} ⟵^{m′} x′.

The relevant length-a mid-parts of this concatenation are the paths from x′ to x_{1} and x_{p} to x_{1} involving m′ and m_{i}, respectively. Hence an entrance to some loop gives the data (a, y) required in Definition 6.16. If the set of base points of loops without entrances has empty interior, then we may find some loop with an entrance with base point in U, so the action is effective. Conversely, assume that U is an open subset so that all points in U are base points of some loop without an entrance. Then there is only one path with range x for any x ∈ U: we must follow the loop based at that point because it has no entrances (note that our paths go backwards). A continuity argument shows that the period of the loop is locally constant. Shrinking U, we may arrange that it is constant equal to p for all x ∈ U. Then mid_{p,a,0} (m) = mid_{0,a,p} (m) for all m ∈ M_{p+a} with r_{p+a} (m) ∈ U. □

If X is discrete, so that we are dealing with an ordinary graph C^{∗}-algebra, then Proposition 6.21 simplifies further to the condition that there are no loops without entrances, which is a standard condition in the theory of graph C^{∗}-algebras (see [66]).

6.3. Gauge-invariant ideals. We now describe which ideals in C^{∗}(H) come from open H-invariant subsets of H^{0} if H is not effective. Recall that the groupoid H is graded by the group completion G of P, H = ⨆ _{g∈G} H_{g}, and that this corresponds to the Fell bundle structure on the Cuntz–Pimsner algebra O, that is, O_{1} = C^{∗}(H_{1}) is the groupoid C^{∗}-algebra of the subgroupoid H_{1}. The canonical projection to O_{1} is a conditional expectation E: O → O_{1}.

Definition 6.22. We call an ideal I ⊆ O gauge-invariant if I is equal to the ideal in O generated by E(I) ⊆ O_{1}.

If G is an Abelian group, so that there is a dual action of Ĝ on O, then standard arguments show that an ideal is “gauge-invariant” in the sense of Definition 6.22 if and only if it is invariant under the dual action of the compact group Ĝ on O in the usual sense because E(x) = ∫Ĝ α_{γ}(x) dγ.

Theorem 6.23. The gauge-invariant ideals in O are in bijection with open invariant subsets of H^{0} or, equivalently, with open, hereditary and saturated subsets of X.

Proof. The bijection between invariant subsets of H^{0} and open, hereditary and saturated subsets of X is Theorem 6.11. An open and invariant subset U ⊆ H^{0} corresponds to the ideal C^{∗}(H|_{U}) ⊆ C^{∗}(H), where H|_{U} denotes the subgroupoid of H with object space U ⊆ H^{0} and arrow space r^{−1} (U) = s^{−1} (U) ⊆ H^{1}. This ideal is gauge-invariant with E(C^{∗}(H|_{U})) = C^{∗}(H_{1}|_{U}). Conversely, let I ⊆ C^{∗}(H) be a gauge-invariant ideal. Then E(I) is an ideal in C^{∗}(H_{1}). The groupoid H_{1} is an approximately proper equivalence relation or a hyperfinite relation. This implies that it is amenable and that any restriction of H_{1} to an invariant closed subset remains effective. Now [9, Corollary 5.9] implies that ideals in C^{∗}(H_{1}) are in bijection with H_{1}-invariant open subsets of H^{0}, where U ⊆ H^{0} corresponds to the ideal C^{∗}(H_{1}|_{U}). This can only be of the form E(I) for an ideal I ⊆ C^{∗}(H) if U is invariant under the whole groupoid H, and then the ideal I generated by C^{∗}(H_{1}|_{U}) is C^{∗}(H|_{U}). Thus any gauge-invariant ideal I is of the form C^{∗}(H|_{U}) for an open H-invariant subset U of H^{0}. □

6.4. Invariant measures. In the following, a “measure” on a locally compact space X means a Radon measure or, equivalently, a positive linear functional on C_{c}(X). We assume X to be second countable and P to be countable, so that H is second countable. Let c: P → (0, ∞) be a homomorphism to the multiplicative group of positive real numbers. This extends to the group completion G and then to H, by letting c|_{H_{p,q}^{1}} = c(p)/c(q). This cocycle on H generates a 1-parameter group of automorphisms of C^{∗}(H), see [69, Section 5]. The one-parameter automorphism groups of C^{∗}(H) described above are not as special as it may seem:

Proposition 6.24. Let H be effective. An automorphism of C^{∗}(H) that acts trivially on the C^{∗}-subalgebra C^{∗}(H_{1}) is given by pointwise multiplication with a homomorphism G → 𝕋. A one-parameter group of such automorphisms comes from a homomorphism c: P → (0, ∞).

Proof. Renault’s construction of a groupoid model for C^{∗}-algebras with a Cartan subalgebra in [72] is natural. More precisely, any automorphism of a twisted groupoid C^{∗}-algebra C^{∗}(H, L) for an étale, effective, Hausdorff, locally compact groupoid H and a Fell line bundle L over H that maps the subalgebra C_{0}(H^{0}) into itself must come from an automorphism of the pair (H, L). The automorphism fixes C_{0}(H^{0}) ⊆ C^{∗}(H_{1}) if and only if the induced automorphism of H acts trivially on objects. For an effective groupoid, this implies that the automorphism acts identically on the inverse semigroup of bisections. Hence it is the identity automorphism. Thus the only source of such automorphisms of C^{∗}(H, L) are automorphisms of the Fell line bundle L; this is the trivial Fell line bundle in our case. Any automorphism of a Fell line bundle over a groupoid H acts by pointwise multiplication with a continuous groupoid homomorphism H → 𝕋. Since we want the automorphism to act identically on C^{∗}(H_{1}), this homomorphism must be constant on H_{1} ⊆ H. Hence it is constant on the subspaces H_{g} and thus comes from a group homomorphism on G. Since G is the group completion of P, homomorphisms G → 𝕋 are in bijection with homomorphisms P → 𝕋. A one-parameter automorphism group is equivalent to a continuous homomorphism P × ℝ → 𝕋; this is equivalent to a continuous homomorphism P → Hom(ℝ, 𝕋) ≅ ℝ ≅ (ℝ_{>0} , ·). Hence all one-parameter automorphism groups of C^{∗}(H) that fix C^{∗}(H_{1}) come from a homomorphism P → (0, ∞). □

Definition 6.25. A measure μ on H^{0} is c-invariant if μ(r(B)) = c(g)μ(s(B)) for any bisection B ⊆ H_{g}^{1}. If c ≡ 1, we speak simply of invariant measures.

A c-invariant measure on H^{0} gives a KMS-weight on C^{∗}(H) for the corresponding automorphism group (with temperature 1); conversely, if H has trivial isotropy groups, then any KMS-weight on C^{∗}(H) for this 1-parameter group of automorphisms is of this form for a unique c-invariant measure on H^{0} (see [69, Proposition 5.4]); this result is generalised by Neshveyev [63] to KMS states on groupoids with non-trivial isotropy, and by Thomsen [75] to KMS weights. In particular, invariant measures on H^{0} give tracial weights on C^{∗}(H); if the set of objects with non-trivial isotropy has measure zero for all invariant measures on H^{0}, then all KMS weights are of this form. We are going to describe invariant measures on H^{0} in terms of measures on X. This requires two operations on measures: push-forwards along continuous maps and pull-backs along local homeomorphisms. The first is standard: if f: X → Y is a continuous map and μ is a measure on X, then f_{∗}μ is the measure on Y defined by f_{∗}μ(B) = μ(f^{−1} (B)) for Borel subsets B ⊆ Y. If f: X → Y is a local homeomorphism and λ is a measure on Y, then f^{∗}λ is the measure on X defined by

f^{∗}λ(B) = ∫_{Y}|f^{−1} (y) ∩ B| dλ(y).

If h: X → ℂ is Borel measurable with compact support, then

∫_{X} h(x) d(f^{∗}λ)(x) = ∫_{Y} ∑_{{x∈X|f(x)=y}} h(x) dλ(y)

because this holds for characteristic functions of Borel subsets.

Definition 6.26. A measure λ on X is c-invariant if λ = c(p)(r_{p})_{∗}s_{p}^{∗}(λ) for all p ∈ P.

Theorem 6.27. The map (π_{1})_{∗} induced by the projection π_{1} : H^{0} → X is a bijection between c-invariant measures on H^{0} and c-invariant measures on X.

Proof. A measure μ on H^{0} gives measures μ_{p} := (π_{p})_{∗}(μ) on M_{p} for all p ∈ P. These are linked by (r_{p,q})_{∗}μ_{pq} = μ_{p} for all p, q ∈ P because r_{p,q} ∘ π_{pq} = π_{p}. Conversely, we claim that any family of measures (μ_{p})_{p∈P} with (r_{p,q})_{∗}μ_{pq} = μ_{p} for all p, q ∈ P comes from a unique measure μ on H^{0}. This is because ⋃ _{p∈P} π_{p}^{∗}(C_{c}(M_{p})) is a dense subspace in C_{c}(H^{0}). The consistency condition r_{p,q} ∘ π_{pq} = π_{p} implies that the positive linear maps C_{c}(M_{p}) → ℂ, f ↦ ∫_{M_{p}} f(x) dμ_{p}(x), well-define a positive linear map on ⋃ _{p∈P} π_{p}^{∗}(C_{c}(M_{p})). This extends uniquely to a positive linear map on C_{c}(H^{0}). Thus we may replace a measure μ on H^{0} by a family of measures (μ_{p})_{p∈P} on the spaces M_{p} whenever this is convenient. Next we claim that μ is c-invariant if and only if μ_{p} = c(p)s_{p}^{∗}μ_{1} for all p ∈ P.

Assume first that μ is c-invariant. Let p ∈ P and let U ⊆ M_{p} be an open, relatively compact subset such that s_{p}|_{U} : U → X is injective. Then

{(x, p, s̃_{p}x) | x ∈ π_{p}^{−1} (U)} ⊆ H_{p,1}^{1}

is a bisection with range π_{p}^{−1} (U) and source π_{1}^{−1} (s_{p}(U)) by Lemma 6.5. Since μ is c-invariant,

μ_{p}(U) = μ(π_{p}^{−1} (U)) = c(p)μ(π_{1}^{−1} (s_{p}(U))) = c(p)μ_{1}(s_{p}(U)).

This equality also holds for all open subsets of U because s_{p} is still injective on them. This implies μ_{p}(B) = c(p)μ_{1}(s_{p}(B)) for all Borel subsets B of U. If B ⊆ M_{p} is an arbitrary Borel subset, then we may cover B by open, relatively compact subsets on which s_{p} is injective because s_{p} is a local homeomorphism. Then we may decompose B as a countable disjoint union B = ⨆ _{i∈ℕ} B_{i} of Borel subsets with B_{i} ⊆ U_{i} for open, relatively compact subsets U_{i} such that s_{p}|_{U_{i}} is injective for all i. Applying the formula above for each i gives

μ_{p}(B) = c(p) ∫_{X} |s_{p}^{−1} (x) ∩ B| dμ_{1}(x) = c(p)s_{p}^{∗}μ_{1}(B).

Thus μ_{p} = c(p)s_{p}^{∗}μ_{1} if μ is c-invariant. Conversely, assume μ_{p} = c(p)s_{p}^{∗}μ_{1}. Let g = pq^{−1} ∈ G and let V ⊆ H_{g}^{1} be a bisection. We may assume without loss of generality that V ⊆ H_{p,q}^{1}, replacing (p, q) by (ph, qh) for some h ∈ P if necessary. Decomposing V into disjoint Borel subsets, we may further reduce to the case where s(V) ⊆ H^{0} is one of the base neighbourhoods in Lemma 6.5; say, s(V) = π_{t}^{−1} (U) for an open subset U ⊆ M_{t}. Replacing (p, q, t) by (pa, qa, tb) for suitable a, b ∈ P, we may arrange t = p by the Ore condition. We assume this from now on. Decomposing V even further, we may arrange that s_{p}|_{U} is injective. Then V is the product of two bisections, one of the form {(x, p, s̃_{p}(x)) | x ∈ π_{p}^{−1} (U)}, the other of the form {(y, q, s̃_{q}(y))^{−1} | y ∈ r(V)}; here s̃_{q}r(V) = s̃_{p}π_{p}^{−1} (U) = π_{1}^{−1} (s_{p}(U)). So r(V) must be of the form π_{q}^{−1} (W) for some W ⊆ M_{q} for which s_{q}|_{W} is a homeomorphism onto s_{p}(U). The upshot of these reductions is that μ is c-invariant once the c-invariance condition holds for bisections of the form {(x, p, s̃_{p}(x)) | x ∈ π_{p}^{−1} (U)} with p ∈ P and an open subset U ⊆ M_{p}. But this is exactly the condition μ_{p} = c(p)s_{p}^{∗}μ_{1}. This finishes the proof of the claim. The claim shows that the family of measures (μ_{p})_{p∈P} and hence the measure μ is determined uniquely by the measure μ_{1} on M_{1} = X provided μ is c-invariant. If we are given a measure λ on X, then μ_{p} := c(p)s_{p}^{∗}(λ) is the only possibility for a c-invariant measure on H^{0} with μ_{1} = λ. This family of measures gives a measure on H^{0} if and only if (r_{p,q})_{∗}μ_{pq} = μ_{p} for all p, q ∈ P. In particular, for p = 1 and q ∈ P, this gives the condition c(q)(r_{q})_{∗}s_{q}^{∗}λ = (r_{q})_{∗}μ_{q} = λ, that is, λ has to be c-invariant. The proof of the theorem will be finished by checking that

c(p)s_{p}^{∗}λ = c(pq)(r_{p,q})_{∗}s_{pq}^{∗}λ

holds for all p, q ∈ P if λ is a c-invariant measure on X. Since c(pq) = c(p)c(q), we have to prove c(q)(r_{p,q})_{∗}s_{pq}^{∗}λ = s_{p}^{∗}λ. Let U ⊆ M_{p} be an open, relatively compact subset. On the one hand, (s_{p}^{∗}λ)(U) := ∫_{X} |s_{p}^{−1} (x) ∩ U| dλ(x). Substituting c(q)(r_{q})_{∗}s_{q}^{∗}λ for λ, this becomes

(6.28) c(q) ∫_{X} |s_{p}^{−1} (x) ∩ U| d(r_{q})_{∗}s_{q}^{∗}λ(x) = c(q) ∫_{M_{q}} |s_{p}^{−1} (r_{q}(z)) ∩ U| ds_{q}^{∗}λ(z)

= c(q) ∫_{X} ∑_{{z∈M_{q}|s_{q}(z)=x}} |s_{p}^{−1} (r_{q}(z)) ∩ U| dλ(x).

On the other hand, (6.29)

c(q)((r_{p,q})_{∗}s_{pq}^{∗}λ)(U) = c(q)s_{pq}^{∗}λ(r_{p,q}^{−1}(U)) = c(q) ∫_{X} |s_{pq}^{−1} (x) ∩ r_{p,q}^{−1}(U)| dλ(x).

We may identify M_{pq} ≅ M_{p} ×_{s_{p} ,X,r_{q}} M_{q} and r_{p,q} with the projection to the first factor. Hence s_{pq}^{−1} (x) ∩ r_{p,q}^{−1}(U) is the set of pairs (y, z) with y ∈ U ⊆ M_{p}, z ∈ M_{q}, s_{p}(y) = r_{q}(z) and s_{q}(z) = x. The cardinality of this set is the sum over all z ∈ s_{q}^{−1} (x) of the cardinalities of s_{p}^{−1} (r_{q}(z)) ∩ U. Hence the right hand sides in (6.28) and (6.29) are equal, as desired. □

Similar arguments as in the end of the proof of Theorem 6.27 show the following:

Remark 6.30. The map α_{p} := (r_{p})_{∗}(s_{p})^{∗} on the space of measures on X is an action of P, that is, α_{pq} = α_{p} ∘ α_{q} for all p, q ∈ P. Therefore, if S ⊆ P generates P, then it suffices to check whether a measure is invariant for p ∈ S.

Example 6.31. Assume that X is discrete. Then a (positive) measure on X is simply a function λ: X → ℝ_{+} . We have (r_{p})_{∗}(s_{p})^{∗}(λ)(x) = ∑ _{r_{p} (m)=x} λ(s_{p}(m)); a c-invariant measure is a non-negative joint eigenvector of these maps with eigenvalue p ↦ c(p). If P = ℕ, then it suffices to look at the generator 1 ∈ ℕ, and the resulting map on the measures of X is matrix-vector multiplication with the adjacency matrix of the graph described by X_{1}. For P = (ℕ^{k}, +), it suffices to look at the k generators of ℕ^{k}. Thus the c-invariant weights on H^{0} are exactly those joint eigenvalues of the adjacency matrices that have a non-negative eigenvector. Any c-invariant weight on H^{0} gives a KMS state on C^{∗}(H) by first applying the conditional expectation C^{∗}(H) → C_{0}(H^{0}) that restricts functions to H^{0} and then integrating against the measure on H^{0}. If all isotropy groups are trivial, then these are all KMS states. Neshveyev [63] shows how to describe KMS states if there is non-trivial isotropy. A particularly simple case is if the set of points in H^{0} with non-trivial isotropy is a null set for all c-invariant measures on H^{0}. In that case, all KMS states for the ℝ-action generated by the cocycle c factor through the conditional expectation, so we get the same answer as if there were no isotropy groups. If the set of non-trivial isotropy has positive measure, then Neshveyev’s description of KMS states uses essentially invariant measurable families of states on the C^{∗}-algebras of the isotropy groups. If there is lots of isotropy, these may be hard to classify. The situation is tractable, however, if the set of points in H^{0} with non-trivial isotropy is countable. This always happens for (separable) graph algebras. The set of orbits of points in H^{0} with non-trivial isotropy is in bijection with simple loops in the graph. The orbit of a point in H^{0} and the set of simple loops are both countable. Thus for a graph C^{∗}-algebra, we may get all KMS states in two steps. First we find the non-negative eigenvectors of the adjacency matrix. Then we examine which loops in the graph give atoms for the induced measure on H^{0}. For each such loop, we take a state on C^{∗}(ℤ), that is, a probability measure on the circle. The KMS states for a non-negative eigenvector of the adjacency matrix are in bijection with the set of all such families of probability measures on the circle. In the case of (higher-rank) graphs, the measures on H^{0} that we get are essentially product measures coming from a measure on the discrete set of vertices V of the graph. If the higher-rank graph is effective, then the set of points with non-trivial isotropy in H^{0} is often also a null set for such product type measures on H^{0}. This explains why the descriptions of the KMS states or weights on (higher-rank) graph C^{∗}-algebras in, for example, [37,38,75], are often equivalent to what we found above, namely, those joint eigenvalues of the k adjacency matrices that have a non-negative eigenvector. As usual, our analysis only applies to regular (higher-rank) graphs.

Remark 6.32. It is shown in [35] that any stably finite, exact, unital C^{∗}-algebra has a tracial state. Conversely, a C^{∗}-algebra with a tracial state is stably finite. Thus a unital, exact C^{∗}-algebra has a tracial state if and only if it is stably finite. We have already remarked that C^{∗}(H) is unital if and only if X′ is compact, and exactness follows if H is amenable, compare Theorem 6.3. Hence Theorem 6.27 gives a necessary and sufficient criterion for C^{∗}(H) to be stably finite in the case where X′ is compact metrisable, P is countable, and H is amenable.

6.5. Local contractivity.

Definition 6.33 ([3]). A locally compact, Hausdorff, étale groupoid G is locally contracting if, for every non-empty open subset U ⊂ G^{0}, there are an open subset V ⊆ U and a bisection B of G such that \overline{V} ⊆ s(B) and r(B\overline{V}) ⊊ V.

Since H is an étale, locally compact groupoid, [3, Proposition 2.4] shows that C_{r}^{∗}(H) is purely infinite (that is, every hereditary C^{∗}-subalgebra contains an infinite projection) if H is essentially free and locally contracting. Actually, we only need H to be effective here by [9, Lemma 3.1.(4)]: this is exactly the condition in [3, Lemma 2.3] that is used in the proof of [3, Proposition 2.4]. We would, however, not expect local contractivity to be necessary for C_{r}^{∗}(H) to be purely infinite. The following definition characterises when the groupoid H is locally contracting in terms of the original action by topological correspondences:

Definition 6.34. An action (M_{p}, σ_{p,q}) of an Ore monoid P on a locally compact Hausdorff space X by proper topological correspondences is locally contracting if for any relatively compact, open subset S ⊆ X′, there are n ∈ ℕ and p_{i}, q_{i}, a_{i}, b_{i} ∈ P and W_{i} ⊆ M′_{p_{i}} ×_{s_{p_{i}} ,X′,s_{q_{i}}} M′_{q_{i}} for 1 ≤ i ≤ n such that

(LC1) p_{1}a_{1} = p_{2}a_{2} = ⋯= p_{n}a_{n} = q_{1}b_{1} = ⋯= q_{n}b_{n}; (LC2) the restricted coordinate projections pr_{1} : W_{i} → M′_{p_{i}} and pr_{2} : W_{i} →

M′_{q_{i}} are injective and open; (LC3) the subsets

pr_{1}(W_{i})M′_{a_{i}} := {σ_{p_{i} ,a_{i}}^{−1} (x_{1}, x_{2}) | x_{1} ∈ pr_{1}(W_{i}), x_{2} ∈ M′_{a_{i}} , s_{p_{i}} (x_{1}) = r_{a_{i}} (x_{2})}

of M′_{p_{i} a_{i}} are disjoint, and so are the subsets pr_{2}(W_{i})M′_{b_{i}};

(LC4) \overline{⨆ _{i=1}^{n} pr_{1}(W_{i})M′_{a_{i}}} ⊊ ⨆ _{i=1}^{n} pr_{2}(W_{i})M′_{b_{i}}; (LC5) r_{q_{i}} pr_{2}(W_{i}) ⊆ S; (LC6) p_{i}q_{i}^{−1} ≠ p_{j} q_{j}^{−1} in G for i ≠ j.

Here X′ ⊆ X is the closed invariant subspace of possible situations, which is different from X if some of the range maps are not surjective.

The choice of a_{i}, b_{i} does not really matter: if the conditions hold for one choice satisfying (LC1), then also for all others. This follows from Lemma 4.16 and the surjectivity of the maps r_{p,q} on the M′_{pq}. Giving up some symmetry, we may use the Ore conditions to simplify the data above slightly: we may assume either p_{1} = p_{2} = ⋯= p_{n} and a_{1} = a_{2} = ⋯= a_{n} or q_{1} = q_{2} = ⋯= q_{n} and b_{1} = b_{2} = ⋯= b_{n}. Condition (LC6) says, roughly speaking, that we cannot make n smaller by combining the data for any i ≠ j. This is its only role, and it could be left out.

Theorem 6.35. The groupoid H is locally contracting if and only if the action (M_{p}, σ_{p,q}) is locally contracting.

Proof. To simplify notation, we replace X by X′ throughout, so that the maps r_{p} are all surjective. Call a subset U of H^{0} good if there are V, B as in Definition 6.33. If U_{1} ⊆ U_{2} and U_{1} is good, then so is U_{2}. If T ⊆ H^{1} is a bisection, then r(T) is good if and only if s(T) is good: given V, B for s(T), then T V and T BT^{−1} will work for r(T). If U ⊆ H^{0} is an arbitrary open subset, then there are p ∈ P and U_{p} ⊆ M_{p} such that π_{p}^{−1} (U_{p}) ⊆ U, s_{p}|_{U_{p}} is injective, and U_{p} is relatively compact and open; here we use that subsets of the form π_{p}^{−1} (U_{p}) for U_{p} ⊆ M_{p} open form a base (Lemma 6.5), that M_{p} is locally compact, and that s_{p} is a local homeomorphism for each p (Lemma 4.10). Hence there is a bisection with range π_{p}^{−1} (U_{p}) and source π_{1}^{−1} (s_{p}(U_{p})). Thus U is good when π_{1}^{−1} (s_{p}(U_{p})) is good. Summing up, all non-empty open subsets of H^{0} are good once all nonempty subsets of the form π_{1}^{−1} (S) with S ⊆ X relatively compact and open are good. Assume now that the action (M_{p}, σ_{p,q}) is locally contracting. Let S ⊂ X be a non-empty, open, relatively compact set. Pick n ∈ ℕ and p_{i}, q_{i}, a_{i}, b_{i} ∈ P and subsets W_{i} ⊆ M_{p_{i}} ×_{s_{p_{i}} ,X,s_{q_{i}}} M_{q_{i}} as in Definition 6.34. Let

B_{i} := {(x_{1}y, p_{i}q_{i}^{−1} , x_{2}y) | (x_{1}, x_{2}) ∈ W_{i}, y ∈ H^{0}, s_{p_{i}} (x_{1}) = π_{1}(y)};

here x_{1}y and x_{2}y are well-defined because s_{p_{i}} (x_{1}) = s_{q_{i}} (x_{2}) = π_{1}(y), which uses that s_{p_{i}} (x_{1}) = s_{q_{i}} (x_{2}) for (x_{1}, x_{2}) ∈ W_{i}. Since s̃_{p_{i}} (x_{1}y) = y = s̃_{q_{i}} (x_{2}y), we have B_{i} ⊆ H_{p_{i} ,q_{i}}^{1} . Condition (LC2) ensures that the range and source maps are open and injective on each B_{i}, so these are bisections. For different i, they have disjoint sources and ranges by (LC3). Hence B := B_{1}⊔B_{2}⊔⋯ ⊔B_{n} is a bisection as well. Condition (LC4) gives \overline{r(B)} ⊊ s(B), and (LC5) gives s(B) ⊆ π_{1}^{−1} (S). Since we assume S to be relatively compact, the closed subset \overline{r(B)} is compact. So there is an open subset V with \overline{r(B)} ⊊ V ⊆ \overline{V} ⊆ s(B). Hence π_{1}^{−1} (S) is good. This implies that H is locally contracting. Conversely, assume that H is locally contracting. Let S ⊆ X be a relatively compact, non-empty, open subset and let U := π_{1}^{−1} (S). Then U is relatively compact, non-empty and open because π_{1} : H^{0} → X is surjective, continuous, and proper. Since H is locally contracting, there is a bisection B ⊆ H^{1} with \overline{r(B)} ⊊ s(B) ⊆ U. Next we have to analyse this bisection B locally. This does not yet use the special feature of B and gives slightly more, namely, a “base” for the inverse semigroup of bisections of H. By this we mean an inverse subsemigroup closed under finite intersections that covers H^{1}. This can be used to study actions of H on C^{∗}-algebras as in [11].

Lemma 6.36. Let B ⊆ H^{1} be a bisection and η ∈ B. Then there is an open bisection B_{η} with η ∈ B_{η} ⊆ B that has the following form:

B_{η} := {(x_{1}y, pq^{−1} , x_{2}y) | (x_{1}, x_{2}) ∈ W, y ∈ H^{0}, s_{p}(x_{1}) = π_{1}(y)}

for some p, q ∈ P and a subset W ⊆ M_{p} ×_{s_{p} ,X,s_{q}} M_{q} such that pr_{1} : W → M_{p} and pr_{2} : W → M_{q} are injective and open. If pt = p_{1}, qt = q_{1} and

W_{1} := {(x_{1}y, x_{2}y) | (x_{1}, x_{2}) ∈ W, y ∈ W_{t}, s_{p}(x_{1}) = r_{t}(y)}

for some t ∈ P, then the data (p, q, W) and (p_{1}, q_{1}, W_{1}) define the same bisection B_{η}.

Proof. We write B_{∗} for the homeomorphism s(B) → r(B) induced by B; this is determined uniquely by B_{∗}(s(h)) = r(h) for all h ∈ B. Let ξ := s(η). There are p, q ∈ P with η ∈ H_{p,q}^{1} because these subsets cover H^{1}. Since the subsets B and H_{p,q}^{1} and the map s: H^{1} → H^{0} are open, s(B ∩ H_{p,q}^{1}) is an open neighbourhood of ξ. Let h ∈ B ∩ H_{p,q}^{1}. Lemma 4.10 shows that there are unique x_{2} ∈ M_{q} and y ∈ H^{0} with s_{q}(x_{2}) = π_{1}(y) and s(h) = x_{2}y, namely, x_{2} = π_{q}(s(h)) and y = s̃_{q}(s(h)). Similarly, there are unique x_{1} ∈ M_{p} and y′ ∈ H^{0} with s_{p}(x_{1}) = π_{1}(y′) and r(h) = x_{1}y′, namely, x_{1} = π_{p}(r(h)) and y′ = s̃_{p}(r(h)). The assumption h ∈ H_{p,q}^{1} means exactly that y = y′. Thus h = (x_{1}y, pq^{−1} , x_{2}y) for x_{1} ∈ M_{p}, x_{2} ∈ M_{q}, y ∈ H^{0} with s_{p}(x_{1}) = s_{q}(x_{2}) = π_{1}(y). Since s_{p} is a local homeomorphism, there is an open neighbourhood V around π_{p}(B_{∗}ξ) ∈ M_{p} so that s_{p}|_{V} : V → X is a homeomorphism onto an open subset of X. The subset

V′:= {x ∈ s(B ∩ H_{p,q}^{1}) | π_{p}(B_{∗}x) ∈ V }

is still an open neighbourhood of ξ. It contains a neighbourhood of ξ that belongs to the base in Lemma 6.5. This gives us q_{2} ∈ P and V′′ ⊂ M_{q_{2}} with ξ ∈ π_{q_{2}}^{−1} (V′′) ⊆ V′. Since s_{q_{2}} is a local homeomorphism as well, we may further shrink V′′so that s_{q_{2}}|_{V′′} becomes injective; we assume this. The first Ore condition gives us a, b ∈ P with qa = q_{2}b. Let p′:= pa and q′:= qa = q_{2}b. Let

W := {(π_{p′}(r(h)), π_{q′}(s(h))) | h ∈ B, π_{q_{2}}(s(h)) ∈ V′′}.

We claim that p′, q′, W have the asserted properties. Let h ∈ B satisfy π_{q_{2}}(s(h)) ∈ V′′. Write s(h) = x_{2}y, r(h) = x_{1}y with x_{1} ∈ M_{p}, x_{2} ∈ M_{q}, y ∈ H^{0} and s_{p}(x_{1}) = s_{q}(x_{2}) = π_{1}(y) as above. Then x_{1} ∈ V, so x_{1} is the unique point in V with s_{p}(x_{1}) = s_{q}(x_{2}). Now write y = y_{1}y_{2} with y_{1} ∈ M_{a}, y_{2} ∈ H^{0}, s_{a}(y_{1}) = π_{1}(y_{2}). Then s(h) = x_{1}y_{1}y_{2} and r(h) = x_{2}y_{1}y_{2}. The point x_{1}y_{1} ∈ M_{pa} = M_{p′} is the unique one in r_{p,a}^{−1}(V) with s_{p,a}(x_{1}y_{1}) = y_{1}. This shows that π_{p′}(r(h)) is determined by π_{q′}(s(h)) and that the map that takes π_{q′}(s(h)) to π_{q′}(r(h)) is continuous. Hence the second coordinate projection pr_{2} : W → M_{q′} is injective and open. Since we assumed s_{q_{2}} to be injective on V′′, the same holds for pr_{1} : W → M_{p′}. If (x_{1}y, pq^{−1} , x_{2}y) ∈ B_{η}, then s̃_{p}(x_{1}y) = y = s̃_{q}(x_{2}y), so (x_{1}y, pq^{−1} , x_{2}y) ∈ H_{p,q}^{1} and B_{η} ⊆ H^{1}. Since pr_{1} : W → M_{p′} and pr_{2} : W → M_{q′} are injective and open, so are the maps s, r: B_{η} → H^{0} because y_{2} is common to both source and range and x_{1}y_{1} and x_{2}y_{1} determine each other uniquely and continuously. Thus B_{η} is a bisection. It is clear from the construction that η ∈ B_{η} ⊆ B. The last statement about different data giving the same B_{η} is implicitly shown above. □

Fix ξ_{0} ∈ s(B) \ \overline{r(B)}. The subset \overline{r(B)} ∪ \{ξ_{0}\} ⊆ s(B) ⊆ U = π_{1}^{−1} (S) is closed and contained in the relatively compact subset π_{1}^{−1} (S), so it is compact. For each x ∈ \overline{r(B)} ∪ \{ξ_{0}\} ⊆ s(B), there is a unique η ∈ B with s(η) = x. Let B_{η} ⊆ B be some bisection as in Lemma 6.36. The open subsets s(B_{η}) for these chosen bisections cover \overline{r(B)} ∪ \{ξ_{0}\}. By compactness, this only needs finitely many of them, say, B_{1}, …, B_{ℓ}. Let B′ := B_{1} ∪ B_{2} … ∪ B_{ℓ} ⊆ B. This is still an open bisection, and it satisfies \overline{r(B′)} ⊆ \overline{r(B)} ⊆ s(B′) ⊆ s(B) ⊆ U. Since ξ_{0} ∈ s(B′) \ \overline{r(B)}, we get \overline{r(B′)} ⊊ s(B′). Hence we may replace B by B′. Each B_{i} is constructed from certain p_{i}, q_{i} ∈ P and W_{i} ⊆ M_{p_{i}} × M_{q_{i}} as in Lemma 6.36. The Ore conditions also provide e, a_{i}, b_{i} ∈ P with p_{i}a_{i} = e = q_{i}b_{i} for all i. Our construction so far already achieves the most crucial properties (LC1), (LC2), (LC4) and (LC5); the last one follows from s(B′) ⊆ s(B) ⊆ π_{1}^{−1} (S). So far, however, the subsets s(B_{i}) and r(B_{i}) may still overlap, and some among the group elements p_{i}q_{i}^{−1} may well be equal. We now rectify this. Let p_{i}q_{i}^{−1} = p_{j} q_{j}^{−1} for some i ≠ j. We may replace (p_{i}, q_{i}) by (p_{i}c, q_{i}c) for c ∈ P using the last statement in Lemma 6.36. By this, we can arrange that p_{i} = p_{j} and q_{i} = q_{j}, which we now assume. Then

B_{i} ∪ B_{j} = {(x_{1}y, pq^{−1} , x_{2}y) | (x_{1}, x_{2}) ∈ W_{i} ∪ W_{j} , y ∈ H^{0}, s_{p}(x_{1}) = π_{1}(y)}.

We know that B_{i} ∪ B_{j} is a bisection because it is contained in the bisection B. Since X = X′, this implies that the coordinate projections remain injective on W_{i} ∪ W_{j}. They are open because this holds locally on W_{i} and W_{j}. Hence we may simply merge the data (p_{i}, q_{i}, W_{i}) and (p_{j} , q_{j} , W_{j}) into one piece, without changing the bisection B′. We go on merging part of our data until all group elements g_{i} := p_{i}q_{i}^{−1} are different. This achieves (LC6). By construction, B_{i} ⊆ H_{p_{i} ,q_{i}}^{1} ⊆ H_{g_{i}}^{1} , and these subsets are disjoint for different g_{i}. Hence the B_{i} are disjoint bisections. Since their union remains a bisection, their ranges and sources must be disjoint, which is (LC3). Hence all the conditions in Definition 6.34 hold. □

Our condition for local contractivity is rather complicated because it is necessary and sufficient. If we restricted to n = 1 in Definition 6.34, the condition would no longer be necessary, and it would depend on the G-grading on H, so it would not be a property of the groupoid H alone. Nevertheless, the case n = 1 of our criterion gives a useful sufficient condition:

Corollary 6.37. The groupoid H is locally contracting if, for any relatively compact, open subset S ⊆ X′, there are p, q, a, b ∈ P with pa = qb and a subset W ⊆ M′_{p} ×_{s_{p} ,X′,s_{q}} M′_{q} such that the projections pr_{1} : W → M′_{p} and pr_{2} : W → M′_{q} are injective and open, r_{q}(pr_{2}(W)) ⊆ S, and \overline{pr_{1}(W)}· M′_{a} ⊊ pr_{2}(W) · M′_{b} as subsets of M′_{pa} = M′_{qb}.

Proof. Besides restricting to n = 1, we also have rewritten

\overline{pr_{1}(W) · M′_{a}}= \overline{pr_{1}(W)}· M′_{a},

using the closure of pr_{1}(W) in M′_{p}. This is because pr_{1}(W) · M′_{a} = pr_{1}(W) ×_{s_{p} ,X′,s_{q}} M′_{a} ⊆ M′_{p} ×_{s_{p} ,X′,s_{q}} M′_{a}, and for such a subset the closure operation clearly works on the first entry only. □

Specialising further, we may assume W to have the form W = W′_{p}×_{s_{p} ,X′,s_{q}} W′_{q} for open subsets W_{p} ⊆ M′_{p} and W_{q} ⊆ M′_{q}; if W has this form, then we may choose W_{p} and W_{q} minimal given W by taking W_{p} = pr_{1}(W) and W_{q} = pr_{2}(W). Then s_{p}(W_{p}) = s_{q}(W_{q}) because s_{p} ∘ pr_{1} = s_{q} ∘ pr_{2} on W.

Lemma 6.38. The maps pr_{1} |_{W} and pr_{2} |_{W} are injective and open if and only if W_{p} ⊆ M′_{p} and W_{q} ⊆ M′_{q} are open and the restrictions s_{p}|_{W_{p}} and s_{q}|_{W_{q}} are injective.

Proof. Assume first that pr_{1} |_{W} and pr_{2} |_{W} are injective and open. Then W_{p} = pr_{1}(W) and W_{q} = pr_{2}(W) are open. If x ∈ W_{p}, then any y ∈ W_{q} with s_{q}(y) = s_{p}(x) gives a point (x, y) ∈ W with pr_{1}(x, y) = x. Hence s_{q}|_{W_{q}} is injective if pr_{1} |_{W} is injective. Conversely, assume that W_{p} ⊆ M′_{p} and W_{q} ⊆ M′_{q} are open subsets and the restrictions s_{p}|_{W_{p}} and s_{q}|_{W_{q}} are injective. Since s_{p} and s_{q} are local homeomorphisms, so are their restrictions to the open subsets W_{p} and W_{q}.

Being also injective and continuous, they are homeomorphisms onto s_{p}(W_{p}) = s_{q}(W_{q}). Hence the coordinate projections on W are homeomorphisms onto W_{p} and W_{q}, respectively. Since these subsets are open, the coordinate projections are injective and open. □

Corollary 6.39. The groupoid H is locally contracting if, for any relatively compact, open subset S ⊆ X′, there are p, q, a, b ∈ P with pa = qb and open subsets W_{p} ⊆ M′_{p} and W_{q} ⊆ M′_{q} such that s_{p}(W_{p}) = s_{q}(W_{q}), the restrictions s_{p}|_{M′_{p}} and s_{q}|_{M′_{q}} are injective, r_{q}(W_{q}) ⊆ S, and \overline{W_{p}} · M′_{a} ⊊ W_{q} · M′_{b} as subsets of M′_{pa} = M′_{qb}.

Proof. Specialise Corollary 6.37 to the case where W = W_{p} ×_{s_{p} ,X′,s_{q}} W_{q}. □

The role of a, b ∈ P is only as padding to be able to compare \overline{W_{p}} and W_{q}. Since we restricted to possible histories, so that all range maps are surjective, we have

\overline{W_{p}} · M′_{a} ⊊ W_{q} · M′_{b} ⟺ \overline{W_{p}} · M′_{at} ⊊ W_{q} · M′_{bt}

for any t ∈ P. This shows that the choice of a, b does not matter: if the criterion holds for one choice, it holds for all choices. The same is true, of course, for Corollary 6.39, and an analogous statement holds for Theorem 6.35. If P is a lattice-ordered Ore monoid, we therefore get equivalent criteria if we take a = p^{−1} (p ∨ q) and b = q^{−1} (p ∨ q) in Corollary 6.37 or Corollary 6.39. We write down the variant of Corollary 6.39:

Corollary 6.40. Assume that P is Ore and lattice-ordered. The groupoid H is locally contracting if, for any relatively compact, open subset S ⊆ X′, there are p, q ∈ P and open subsets W_{p} ⊆ M′_{p} and W_{q} ⊆ M′_{q} such that s_{p}(W_{p}) = s_{q}(W_{q}), the restrictions s_{p}|_{M′_{p}} and s_{q}|_{M′_{q}} are injective, r_{q}(W_{q}) ⊆ S, and \overline{W_{p}} · M′_{a} ⊊ W_{q} · M′_{b} as subsets of M′_{pa} = M′_{qb}, where a = p^{−1} (p ∨ q) and b = q^{−1} (p ∨ q).

The inclusion \overline{W_{p}}· M′_{a} ⊆ W_{q} · M′_{b} for a = p^{−1} (p ∨ q) and b = q^{−1} (p ∨ q) means that for any x_{p} ∈ \overline{W_{p}} there is x_{q} ∈ W_{q} so that x_{p} and x_{q} have a common extension; the minimal such extension lives in M′_{pa} = M′_{qb}. The meaning of \overline{W_{p}} · M′_{a} ≠ W_{q} · M′_{b} is that some y ∈ W_{q} has no common extension with any element of \overline{M_{p}}. Corollary 6.40 involves the same ingredients as the sufficient condition for the boundary path groupoid of a higher-rank topological graph to be locally contracting in [73, Proposition 5.8]. The proof of [73, Lemma 5.9] shows that our sufficient criterion is more general than the one in [73]. We are, however, not aware of any applications of the criterion in [73]. Our criteria have the advantage that we understand very well which assumptions we have imposed on the contracting bisections for H. In Corollary 6.37, the only assumption is that the bisection is contained in H_{g}^{1} for some g ∈ G. In Corollaries 6.39 and 6.40, we assume this and that the bisection has a product form.

1398 Suliman Albandik and Ralf Meyer

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Suliman Albandik Mathematisches Institut Universität Göttingen Bunsenstraße 3–5 37073 Göttingen Germany albandik@uni-goettingen.de Ralf Meyer Mathematisches Institut Universität Göttingen Bunsenstraße 3–5 37073 Göttingen Germany rmeyer2@uni-goettingen.de