C*-Algebras of Boolean Inverse Monoids –

Traces and Invariant Means

Charles Starling^{1}

Received: May 19, 2016 Revised: June 21, 2016

Communicated by Joachim Cuntz

Abstract. To a Boolean inverse monoid S we associate a universal C*-algebra C_{B}^{∗}(S) and show that it is equal to Exel’s tight C*-algebra of S. We then show that any invariant mean on S (in the sense of Kudryavtseva, Lawson, Lenz and Resende) gives rise to a trace on C_{B}^{∗}(S), and vice-versa, under a condition on S equivalent to the underlying groupoid being Hausdorff. Under certain mild conditions, the space of traces of C_{B}^{∗}(S) is shown to be isomorphic to the space of invariant means of S. We then use many known results about traces of C*-algebras to draw conclusions about invariant means on Boolean inverse monoids; in particular we quote a result of Blackadar to show that any metrizable Choquet simplex arises as the space of invariant means for some AF inverse monoid S.

2010 Mathematics Subject Classification: 20M18, 46L55, 46L05

1 Introduction

This article is the continuation of our study of the relationship between inverse semigroups and C*-algebras. An inverse semigroup is a semigroup S for which every element s ∈ S has a unique “inverse” s^{∗} in the sense that

ss^{∗}s = s and s^{∗}ss^{∗} = s^{∗}.

An important subsemigroup of any inverse semigroup is its set of idempotents E(S) = {e ∈ S | e^{2} = e} = {s^{∗}s | s ∈ S}. Any set of partial isometries closed under product and involution inside a C*-algebra is an inverse semigroup, and its set of idempotents forms a commuting set of projections. Many C*-algebras A have been profitably studied in the following way:

1Supported by the NSERC grants of Benoît Collins, Thierry Giordano, and Vladimir Pestov. cstar050@uottawa.ca.

Documenta Mathematica 21 (2016) 809–840

1. identify a generating inverse semigroup S,

2. write down an abstract characterization of S,

3. show that A is universal for some class of representations of S.

We say “some class” above because typically considering all representations (as in the construction of Paterson [Pat99]) gives us a larger C*-algebra than we started with. For example, consider the multiplicative semigroup inside the Cuntz algebra O_{2} generated by the two canonical generators s_{0} and s_{1}; in semigroup literature this is usually denoted P_{2} and called the polycyclic monoid of order 2. The C*-algebra which is universal for all representations of P_{2} is T_{2}, the Toeplitz extension of O_{2}. In an effort to arrive back at the original C*- algebra in cases such as this, Exel defined the notion of tight representations [Exe08], and showed that the universal C*-algebras for tight representations of P_{2} is O_{2}. See [Sta16], [Sta15], [EP16], [EP14], [EGS12], [COP15] for other examples of this approach. Another approach to this issue is to instead alter the inverse semigroup S. An inverse semigroup carries with it a natural order structure, and when an inverse semigroup S is represented in a C*-algebra A, two elements s, t ∈ S, which did not have a lowest upper bound in S, may have one inside A. So, from P_{2}, Lawson and Scott [LS14, Proposition 3.32] constructed a new inverse semigroup C_{2}, called the Cuntz inverse monoid, by adding to P_{2} all possible joins of compatible elements (s, t are compatible if s^{∗}t, st^{∗} ∈ E(S)). The Cuntz inverse monoid is an example of a Boolean inverse monoid, and the goal of this paper is to define universal C*-algebras for such monoids and study them. A Boolean inverse monoid is an inverse semigroup which contains joins of all finite compatible sets of elements and whose idempotent set is a Boolean algebra. To properly represent a Boolean inverse monoid S, one reasons, one should insist that the join of two compatible s, t ∈ S be sent to the join of the images of s and t. We prove in Proposition 3.3 that such a representation is necessarily a tight representation, and so we obtain that the universal C*- algebra of a Boolean inverse monoid (which we denote C_{B}^{∗}(S)) is exactly its tight C*-algebra, Theorem 3.5. This is the starting point of our study, as the universal tight C*-algebra can be realized as the C*-algebra of an ample groupoid. The main inspiration of this paper is [KLLR16] which defines and studies invariant means on Boolean inverse monoids. An invariant mean is a function μ : E(S) → [0, ∞) such that μ(e∨f) = μ(e)+μ(f) when e and f are orthogonal, and such that μ(ss^{∗}) = μ(s^{∗}s) for all s ∈ S. If one thinks of the idempotents as clopen sets in the Stone space of the Boolean algebra E(S), such a function has the flavour of an invariant measure or a trace. We make this precise in Section 4: as long as S satisfies a condition which guarantees that the induced groupoid is Hausdorff (which we call condition (H)), every invariant mean on S gives rise to a trace on C_{B}^{∗}(S) (Proposition 4.6) and every trace on C_{B}^{∗}(S) gives rise to an invariant mean on S (Proposition 4.7). This becomes a oneto-one correspondence if we assume that the associated groupoid G_{tight}(S) is principal and amenable (Theorem 4.13). We also prove that, whether G_{tight}(S) is principal and amenable or not, there is an affine isomorphism between the space of invariant means on S and the space of G_{tight}(S)-invariant measures on its unit space (Proposition 4.11). In the final section, we apply our results to examples of interest. We study the AF inverse monoids in detail – these are Boolean inverse monoids arising from Bratteli diagrams in much the same way as AF C*-algebras. As it should be, given a Bratteli diagram, the C*-algebra of its Boolean inverse monoid is isomorphic to the AF algebra it determines (Theorem 5.1). From this we can conclude, using the results of Section 4 and the seminal result of Blackadar [Bla80], that any Choquet simplex arises as the space of invariant means for some Boolean inverse monoid. We go on to consider two examples where there is typically only one invariant mean, those being self-similar groups and aperiodic tilings.

2 Preliminaries and notation

We will use the following general notation. If X is a set and U ⊂ X, let Id_{U} denote the map from U to U which fixes every point, and let 1_{U} denote the characteristic function on U, i.e. 1_{U} : X → ℂ defined by 1_{U}(x) = 1 if x ∈ U and 1_{U}(x) = 0 if x ∉ U. If F is a finite subset of X, we write F ⊂_{fin} X.

2.1 Inverse semigroups

An inverse semigroup is a semigroup S such that for all s ∈ S, there is a unique element s^{∗} ∈ S such that

ss^{∗}s = s, s^{∗}ss^{∗} = s^{∗}.

The element s^{∗} is called the inverse of s. All inverse semigroups in this paper are assumed to be discrete and countable. For s, t ∈ S, one has (s^{∗})^{∗} = s and (st)^{∗} = t^{∗}s^{∗}. Although not implied by the definition, we will always assume that inverse semigroups have a 0 element, that is, an element such that

0s = s0 = 0 for all s ∈ S.

An inverse semigroup with identity is called an inverse monoid. Even though we call s^{∗} the inverse of s, we need not have ss^{∗} = 1, although it is always true that (ss^{∗})^{2} = ss^{∗}ss^{∗} = ss^{∗}, i.e. ss^{∗} (and s^{∗}s for that matter) is an idempotent. We denote the set of all idempotents in S by

E(S) = {e ∈ S | e^{2} = e}.

It is a nontrivial fact that if S is an inverse semigroup, then E(S) is closed under multiplication and commutative. It is also clear that if e ∈ E(S), then e^{∗} = e.

Let X be a set, and let

I(X) = {f : U → V | U, V ⊂ X, f bijective}.

Then I(X) is an inverse monoid with the operation of composition on the largest possible domain, and inverse given by function inverse; this is called the symmetric inverse monoid on X. Every idempotent in I(X) is given by Id_{U} for some U ⊂ X. The function Id_{X} is the identity for I(X), and the empty function is the 0 element for I(X). The fundamental Wagner-Preston theorem states that every inverse semigroup is embeddable in I(X) for some set X – one can think of this as analogous to the Cayley theorem for groups. Every inverse semigroup carries a natural order structure: for s, t ∈ S we say s ⩽ t if and only if ts^{∗}s = s, which is also equivalent to ss^{∗}t = s. For elements e, f ∈ E(S), we have e ⩽ f if and only if ef = e. As usual, for s, t ∈ S, the join (or least upper bound) of s and t will be denoted s ∨ t (if it exists), and the meet (or greatest lower bound) of s and t will be denoted s ∧ t (if it exists). For A ⊂ S, we let A^{↑} = {t ∈ S | s ⩽ t for some s ∈ A} and A^{↓} = {t ∈ S | t ⩽ s for some s ∈ A}. If s, t ∈ S, then we say s and t are compatible if s^{∗}t, st^{∗} ∈ E(S), and a set F ⊂ S is called compatible if all pairs of elements of F are compatible.

Definition 2.1. An inverse semigroup S is called distributive if whenever we have a compatible set F ⊂_{fin} S, then ⋁_{s∈F} s exists in S, and for all t ∈ S we have

t ( ⋁_{s∈F} s) = ⋁_{s∈F} ts  and  ( ⋁_{s∈F} s) t = ⋁_{s∈F} st.

In the natural partial order, the idempotents form a meet semilattice, which is to say that any two elements e, f ∈ E(S) have a meet, namely ef. If C ⊂ X ⊂ E(S), we say that C is a cover of X if for all x ∈ X there exists c ∈ C such that cx ≠ 0. In a distributive inverse semigroup each pair of idempotents has a join in addition to the meet mentioned above, but in general E(S) will not have relative complements and so in general will not be a Boolean algebra. The case where E(S) is a Boolean algebra is the subject of the present paper.

Definition 2.2. A Boolean inverse monoid is a distributive inverse monoid S with the property that E(S) is a Boolean algebra, that is, for every e ∈ E(S) there exists e^{⊥} ∈ E(S) such that ee^{⊥} = 0, e ∨ e^{⊥} = 1, and the operations ∨, ∧, ⊥ satisfy the laws of a Boolean algebra [GH09, Chapter 2].

Example 2.3. Perhaps the best way to think about the order structure and related concepts above is by describing them on I(X), which turns out to be a Boolean inverse monoid. Firstly, for g, h ∈ I(X), g ⩽ h if and only if h extends g as a function. In I(X), two functions f and g are compatible if they agree on the intersection of their domains and their inverses agree on the intersection of their ranges. In such a situation, one can form the join f ∨ g which is the union of the two functions; this will again be an element of I(X). Composing h ∈ I(X) with f ∨ g will be the same as hf ∨ hg. Finally, E(I(X)) = {Id_{U} | U ⊂ X} is a Boolean algebra (isomorphic to the Boolean algebra of all subsets of X) with Id_{U}^{⊥} = Id_{U^{c}}.

2.2 Étale groupoids

A groupoid is a small category where every arrow is invertible. If G is a groupoid, the set of elements γγ^{−1} is denoted G^{(0)} and is called the set of units of G. The maps r : G → G^{(0)} and d : G → G^{(0)} defined by r(γ) = γγ^{−1} and d(γ) = γ^{−1}γ are called the range and source maps, respectively. The set G^{(2)} = {(γ, η) ∈ G^{2} | r(η) = d(γ)} is called the set of composable pairs. A topological groupoid is a groupoid G which is a topological space and for which the inverse map from G to G and the product from G^{(2)} to G are both continuous (where in the latter, the topology on G^{(2)} is the product topology inherited from G^{2}). We say that a topological groupoid G is étale if it is locally compact, second countable, G^{(0)} is Hausdorff, and the maps r and d are both local homeomorphisms. Note that an étale groupoid need not be Hausdorff. If G is étale, then G^{(0)} is open, and G is Hausdorff if and only if G^{(0)} is closed (see for example [EP16, Proposition 3.10]). For x ∈ G^{(0)}, let G(x) = {γ ∈ G | r(γ) = d(γ) = x} – this is a group, and is called the isotropy group at x. A groupoid G is said to be principal if all the isotropy groups are trivial, and a topological groupoid is said to be essentially principal if the points with trivial isotropy groups are dense in G^{(0)}. A topological groupoid is said to be minimal if for all x ∈ G^{(0)}, the set O_{G}(x) = r(d^{−1}(x)) is dense in G^{(0)} (the set O_{G}(x) is called the orbit of x). If G is an étale groupoid, an open set U ⊂ G is called a bisection if r|_{U} and d|_{U} are both injective (and hence homeomorphisms). The set of all bisections is denoted G^{op} and is a distributive inverse semigroup when given the operations of setwise product and inverse. We say that an étale groupoid G is ample if the set of compact bisections forms a basis for the topology on G. The set of compact bisections is called the ample semigroup of G, is denoted G^{a}, and is also a distributive inverse subsemigroup of G^{op} [LL13, Lemma 3.14]. Since G is second countable, G^{a} must be countable [Exe10, Corollary 4.3]. If G^{(0)} is compact, then the idempotent set of G^{a} is the set of all clopen sets in G^{(0)}, and so G^{a} is a Boolean inverse monoid (see also [Ste10, Proposition 3.7] which shows that when G is Hausdorff and G^{(0)} is only locally compact, G^{a} is a Boolean inverse semigroup, i.e. a distributive inverse semigroup whose idempotent semilattice is a generalized Boolean algebra). To an étale groupoid G one can associate C*-algebras through the theory developed by Renault [Ren80]. Let C_{c}(G) denote the linear space of continuous compactly supported functions on G. Then C_{c}(G) becomes a ∗-algebra with product and involution given by

fg(γ) = ∑_{γ_{1}γ_{2}=γ} f(γ_{1})g(γ_{2}), f^{∗}(γ) = \overline{f(γ^{−1})}.

From this one can produce two C*-algebras C^{∗}(G) and C_{red}^{∗}(G) (called the C*-algebra of G and the reduced C*-algebra of G, respectively) by completing C_{c}(G) in certain norms, see [Ren80, Definitions 1.12 and 2.8]. There is always a surjective ∗-homomorphism Λ : C^{∗}(G) → C_{red}^{∗}(G), and if Λ is an isomorphism we say that G satisfies weak containment. If G is amenable [ADR00], then G satisfies weak containment. There is an example of a case where Λ is an isomorphism for a nonamenable groupoid [Wil15], but under some conditions on G one has that weak containment and amenability are equivalent, see [AD16b, Theorem B]. Recall that if B ⊂ A are both C*-algebras, then a surjective linear map E : A → B is called a conditional expectation if E is contractive, E ∘ E = E, and E(bac) = bE(a)c for all b, c ∈ B and a ∈ A. Let G be a Hausdorff étale groupoid with compact unit space, and consider the map E : C_{c}(G) → C(G^{(0)}) defined by

E(f) = f|_{G^{(0)}} .  (1)

Then this map extends to a conditional expectation on both C^{∗}(G) and C_{red}^{∗}(G), both denoted E. On C_{red}^{∗}(G), E is faithful in the sense that if E(a^{∗}a) = 0, then a = 0. Let G be an ample étale groupoid. Both C*-algebras contain C_{c}(G), and hence if U is a compact bisection, 1_{U} is an element of both C*-algebras. Hence we have a map π : G^{a} → C^{∗}(G) given by π(U) = 1_{U}. This map satisfies π(UV) = π(U)π(V), π(U^{−1}), and π(0) = 0, in other words, π is a representation of the inverse semigroup G^{a} [Exe10].

2.3 The tight groupoid of an inverse semigroup

Let S be an inverse semigroup. A filter in E(S) is a nonempty subset ξ ⊂ E(S) such that

1. 0 ∉ ξ,

2. e, f ∈ ξ implies that ef ∈ ξ, and

3. e ∈ ξ, e ⩽ f implies f ∈ ξ.

The set of filters is denoted Ê_{0}(S), and can be viewed as a subspace of {0, 1}^{E(S)}. For X, Y ⊂_{fin} E(S), let

U(X, Y) = {ξ ∈ Ê_{0}(S) | X ⊂ ξ, Y ∩ ξ = ∅}.

sets of this form are clopen and generate the topology on Ê_{0}(S) as X and Y vary over all the finite subsets of E(S). With this topology, Ê_{0}(S) is called the spectrum of E(S).

A filter is called an ultrafilter if it is not properly contained in any other filter. The set of all ultrafilters is denoted Ê_{∞}(S). As a subspace of Ê_{0}(S), Ê_{∞}(S) may not be closed. Let Ê_{tight}(S) denote the closure of Ê_{∞}(S) in Ê_{0}(S) – this is called the tight spectrum of E(S). Of course, when E(S) is a Boolean algebra, Ê_{tight}(S) = Ê_{∞}(S) by Stone duality [GH09, Chapter 34]. An action of an inverse semigroup S on a locally compact space X is a semigroup homomorphism α : S → I(X) such that

1. α_{s} is continuous for all s ∈ S,

2. the domain of α_{s} is open for each s ∈ S, and

3. the union of the domains of the α_{s} is equal to X.

If α is an action of S on X, we write α : S ↷ X. The above implies that α_{s^{∗}} = α_{s}^{−1}, and so each α_{s} is a homeomorphism. For each e ∈ E(S), the map α_{e} is the identity on some open subset D_{e}^{α}, and one easily sees that the domain of α_{s} is D_{s^{∗}s}^{α} and the range of α_{s} is D_{ss^{∗}}^{α}, that is

α_{s} : D_{s^{∗}s}^{α} → D_{ss^{∗}}^{α}.

There is a natural action θ of S on Ê_{tight}(S); this is referred to in [EP16] as the standard action of S. For e ∈ E(S), let D_{e}^{θ} = {ξ ∈ Ê_{tight}(S) | e ∈ ξ} = U({e}, ∅) ∩ Ê_{tight}(S). For each s ∈ S and ξ ∈ D_{s^{∗}s}^{θ}, define θ_{s}(ξ) = {ses^{∗} | e ∈ ξ}^{↑} – this is a well-defined homeomorphism from D_{s^{∗}s}^{θ} to D_{ss^{∗}}^{θ}, for the details, see [Exe08]. One can associate a groupoid to an action α : S ↷ X. Let S ×_{α} X = {(s, x) ∈ S × X | x ∈ D_{s^{∗}s}^{α}}, and put an equivalence relation ∼ on this set by saying that (s, x) ∼ (t, y) if and only if x = y and there exists some e ∈ E(S) such that se = te and x ∈ D_{e}^{α}. The set of equivalence classes is denoted

G(α) = {[s, x] | s ∈ S, x ∈ X}

and becomes a groupoid when given the operations

d([s, x]) = x, r([s, x]) = α_{s}(x),
[s, x]^{−1} = [s^{∗}, α_{s}(x)], [t, α_{s}(x)][s, x] = [ts, x].

This is called the groupoid of germs of α. Note that above we are making the identification of the unit space with X, because [e, x] = [f, x] for any e, f ∈ E(S) with x ∈ D_{e}^{α}, D_{f}^{α}. For s ∈ S and open set U ⊂ D_{s^{∗}s}^{α} we let

Θ(s, U) = {[s, x] | x ∈ U}

and endow G(α) with the topology generated by such sets. With this topology G(α) is an étale groupoid, sets of the above type are bisections, and if X is totally disconnected G(α) is ample.

Let θ : S ↷ Ê_{tight}(S) be the standard action, and define

G_{tight}(S) = G(θ).

This is called the tight groupoid of S. This was defined first in [Exe08] and studied extensively in [EP16]. Let G be an ample étale groupoid, and consider the Boolean inverse monoid G^{a}. By work of Exel [Exe10] if one uses the above procedure to produce a groupoid from G^{a}, one ends up with exactly G. In symbols,

G_{tight}(G^{a}) ≅ G for any ample étale groupoid G.  (2)

We note this result was also obtained in [Len08, Theorem 6.11] in the case where Ê_{tight}(S) = Ê_{∞}(S). In particular,

G_{tight}(G_{tight}(S)^{a}) ≅ G_{tight}(S) for all inverse semigroups S.

This result can be made categorical [LL13, Theorem 3.26], and has been generalized to cases where the space of units is not even Hausdorff. This duality between Boolean inverse semigroups and ample étale groupoids falls under the broader program of noncommutative Stone duality, see [LL13] for more details.

3 C*-algebras of Boolean inverse monoids

In this section we describe the tight C*-algebra of a general inverse monoid, define the C*-algebra of a Boolean inverse monoid, and show that these two notions coincide for Boolean inverse monoids. If S is an inverse monoid, then a representation of S in a unital C*-algebra A is a map π : S → A such that π(0) = 0, π(s^{∗}) = π(s)^{∗}, and π(st) = π(s)π(t) for all s, t ∈ S. If π is a representation, then C^{∗}(π(E(S))) is a commutative C*-algebra. Let

ℬ_{π} = {e ∈ C^{∗}(π(E(S))) | e^{2} = e = e^{∗}}

Then this set is a Boolean algebra with operations

e ∧ f = ef,  e ∨ f = e + f − ef,  e^{⊥} = 1 − e.

We will be interested in a subclass of representations of S. Take X, Y ⊂_{fin} E(S), and define

E(S)^{X,Y} = {e ∈ E(S) | e ⩽ x for all x ∈ X, ey = 0 for all y ∈ Y }

We say that a representation π : S → A with A unital is tight if for all X, Y, Z ⊂_{fin} E(S) where Z is a cover of E(S)^{X,Y}, we have the equation

⋁_{z∈Z} π(z) = ∏_{x∈X} π(x) ∏_{y∈Y} (1 − π(y)).  (3)

The tight C*-algebra of S, denoted C_{tight}^{∗}(S), is then the universal unital C*- algebra generated by one element for each element of S subject to the relations that guarantee that the standard map from S to C_{tight}^{∗}(S) is tight. The above was all defined in [Exe08] and the interested reader is directed there for the details. It is a fact that C_{tight}^{∗}(S) ≅ C^{∗}(G_{tight}(S)) where the latter is the full groupoid C*-algebra (see e.g. [Exe10, Theorem 2.4]). If S has the additional structure of being a Boolean inverse monoid, then we might wonder what extra properties π should have, in particular, what is the notion of a “join” of two partial isometries in a C*-algebra? Let A be a C*-algebra, and suppose that S is a Boolean inverse monoid of partial isometries in A. If we have s, t ∈ S such that s^{∗}t, st^{∗} ∈ E(S), then

tt^{∗}s = tt^{∗}ss^{∗}s = ss^{∗}tt^{∗}s = s(s^{∗}t)(s^{∗}t)^{∗} = ss^{∗}t

and if we let a_{s,t} := s + t − ss^{∗}t = s + t − tt^{∗}s, this is a partial isometry with range a_{ss^{∗},tt^{∗}} and support a_{s^{∗}s,t^{∗}t}. A short calculation shows that a_{s,t} is the least upper bound for s and t in the natural partial order, and so a_{s,t} = s ∨ t. It is also straightforward that r(s ∨ t) = rs ∨ rt for all r, s, t ∈ S. This leads us to the following definitions.

Definition 3.1. Let S be a Boolean inverse monoid. A Boolean inverse monoid representation of S in a unital C*-algebra A is a map π : S → A such that

1. π(0) = 0,

2. π(st) = π(s)π(t) for all s, t ∈ S,

3. π(s^{∗}) = π(s)^{∗} for all s ∈ S, and

4. π(s ∨ t) = π(s) + π(t) − π(ss^{∗}t) for all compatible s, t, ∈ S.

Definition 3.2. Let S be a Boolean inverse monoid. Then the universal C*- algebra of S, denoted C_{B}^{∗}(S), is defined to be the universal unital C*-algebra generated by one element for each element of S subject to the relations which say that the standard map of S into C_{B}^{∗}(S) is a Boolean inverse monoid representation. The map π_{u} which takes an element s to its corresponding element in C_{B}^{∗}(S) will be called the universal Boolean inverse monoid representation of S, and we will sometimes use the notation δ_{s} := π_{u}(s).

The theory of tight representations was originally developed to deal with representing inverse semigroups (in which joins may not exist) inside C*-algebras, because in a C*-algebra two commuting projections always have a join. It should come as no surprise then that once we are dealing with an inverse semigroup where we \underline{can} take joins, the representations which respect joins end up being exactly the tight representations, see [DM14, Corollary 2.3]. This is what we prove in the next proposition.

Proposition 3.3. Let S be a Boolean inverse monoid. Then a map π : S → A is a Boolean inverse monoid representation of S if and only if π is a tight representation.

Proof. Suppose that π is a Boolean inverse monoid representation of S. Then when restricted to E(S), π is a Boolean algebra homomorphism into ℬ_{π}, and so by [Exe08, Proposition 11.9], π is a tight representation. On the other hand, suppose that π is a tight representation, and first suppose that e, f ∈ E(S). Then the set {e, f} is a cover for E(S)^{{e∨f},∅}, so

π(e) ∨ π(f) = π(e ∨ f).

Now let s, t ∈ S be compatible, so that s^{∗}t = t^{∗}s and st^{∗} = ts^{∗} are both idempotents, and we have

s^{∗}st^{∗}t = s^{∗}ts^{∗}t = s^{∗}t.

Since (s ∨ t)^{∗}(s ∨ t) = s^{∗}s ∨ t^{∗}t, we have

π(s ∨ t) = π(s ∨ t)π(s^{∗}s ∨ t^{∗}t)
= π(s ∨ t)(π(s^{∗}s) + π(t ∗ t) − π(s^{∗}st^{∗}t)
= π(ss^{∗}s ∨ ts^{∗}s) + π(st^{∗}t ∨ tt^{∗}t) − π(ss^{∗}st^{∗}t ∨ tt^{∗}ts^{∗}s))
= π(s ∨ st^{∗}s) + π(ts^{∗}t ∨ t) − π(st^{∗}t ∨ ts^{∗}s)
= π(s) + π(t) − π(ss^{∗}t)

where the last line follows from the facts that st^{∗}s ⩽ s, ts^{∗}t ⩽ t and ts^{∗}s = st^{∗}t = ss^{∗}t = tt^{∗}s.

We have the following consequence of the proof of the above proposition.

Corollary 3.4. Let S be a Boolean inverse monoid. Then a map π : S → A is a Boolean inverse monoid representation of S if and only if it is a representation and for all e, f ∈ E(S) we have π(e ∨ f) = π(e) + π(f) − π(ef).

We now have the following.

Theorem 3.5. Let S be a Boolean inverse monoid. Then

C_{B}^{∗}(S) ≅ C_{tight}^{∗}(S) ≅ C^{∗}(G_{tight}(S)).

In what follows, we will be studying traces on C*-algebras arising from Boolean inverse monoids. However, many of our examples will actually arise from inverse monoids which are not distributive, and so the Boolean inverse monoid in question will actually be G_{tight}(S)^{a}, see (2). The map from S to G_{tight}(S)^{a} defined by

s ↦ Θ(s, D_{s^{∗}s}^{θ})

may fail to be injective, and so we cannot say that a given inverse monoid can be embedded in a Boolean inverse monoid. The obstruction arises from the following situation: suppose S is an inverse semigroup and that we have e, f ∈ E(S) such that e ⩽ f and for all 0 ≠ k ⩽ f we have ek ≠ 0, in other words, {e} is a cover for {f}^{↓}. In such a situation, we say that e is dense in f^{2}, and by (3) we must have that π(e) = π(f) (see also [Exe09] and [Exe08, Proposition 11.11]). For most of our examples, we will be considering inverse semigroups which have faithful tight representations, though we consider one which does not. We close this section by recording some consequences of Theorem 3.5. The tight groupoid and tight C*-algebra of an inverse semigroup were extensively studied in [EP16] and [Ste16], where they gave conditions on S which imply that C_{tight}^{∗}(S) is simple and purely infinite. We first recall some definitions from [EP16].

Definition 3.6. Let S be an inverse semigroup, let s ∈ S and e ⩽ s^{∗}s. Then we say that

1. e is fixed by s if se = e, and

2. e is weakly fixed by s if for all 0 ≠ f ⩽ e, fsfs^{∗} ≠ 0.

Denote by J_{s} := {e ∈ E(S) | se = e} the set of all fixed idempotents for s ∈ S. We note that an inverse semigroup for which J_{s} = {0} for all s ∉ E(S) is called E*-unitary.

Theorem 3.7. Let S be an inverse semigroup. Then

1. G_{tight}(S) is Hausdorff if and only if J_{s} has a finite cover for all s ∈ S. [EP16, Theorem 3.16]

2. If G_{tight}(S) is Hausdorff, then G_{tight}(S) is essentially principal if and only if for every s ∈ S and every e ∈ E(S) weakly fixed by s, there exists a finite cover for {e} by fixed idempotents. [EP16, Theorem 4.10]

3. G_{tight}(S) is minimal if and only if for every nonzero e, f ∈ E(S), there exist F ⊂_{fin} S such that {esfs^{∗} | s ∈ F } is a cover for {e}.[EP16, Theorem 5.5]

We translate the above to the case where S is a Boolean inverse monoid.

Proposition 3.8. Let S be a Boolean inverse monoid. Then

1. G_{tight}(S) is Hausdorff if and only if for all s ∈ S, there exists an idempotent e_{s} with se_{s} = e_{s} such that if e is fixed by s, then e ⩽ e_{s}.

2. If G_{tight}(S) is Hausdorff, then G_{tight}(S) is essentially principal if and only if for every s ∈ S, e weakly fixed by s implies e is fixed by s.

3. G_{tight}(S) is minimal if and only if for every nonzero e, f ∈ E(S), there exist F ⊂_{fin} S such that e ⩽ ⋁_{s∈F} sfs^{∗}.

2This is the terminology used in [Exe08, Definition 11.10] and [Exe09], though in [LS14, Section 6.3] such an e is called essential in f.

Documenta Mathematica 21 (2016) 809–840

Proof. Statements 2 and 3 are easy consequences of taking the joins of the finite covers mentioned. Statement 1 is central to what follows, and is proven in Lemma 4.2.

If an étale groupoid G is Hausdorff, then C^{∗}(G) is simple if and only if G is essentially principal, minimal, and satisfies weak containment, see [BCFS14] (also see [ES15] for a discussion of amenability of groupoids associated to inverse semigroups).

4 Invariant means and traces

In this section we consider invariant means on Boolean inverse monoids, and show that such functions always give rise to traces on the associated C*- algebras. This definition is from [KLLR16].

Definition 4.1. Let S be a Boolean inverse monoid. A nonzero function μ : E(S) → [0, ∞) will be called an invariant mean if

1. μ(s^{∗}s) = μ(ss^{∗}) for all s ∈ S

2. μ(e ∨ f) = μ(e) + μ(f) for all e, f ∈ E(S) such that ef = 0.

If in addition μ(1) = 1, we call μ a normalized invariant mean. An invariant mean μ will be called faithful if μ(e) = 0 implies e = 0. We will denote by M(S) the affine space of all normalized invariant means on S.

We make an important assumption on the Boolean inverse monoids we consider here. This assumption is equivalent to the groupoid G_{tight}(S) being Hausdorff [EP16, Theorem 3.16].^{3}

For every s ∈ S, the set J_{s} = {e ∈ E(S) | se = e} admits a finite cover. (H)

The next lemma records straightforward consequences of condition (H) when S happens to be a Boolean inverse monoid.

Lemma 4.2. Let S be Boolean inverse monoid which satisfies condition (H). Then,

1. for each s ∈ S there is an idempotent e_{s} such that for any finite cover C

of J_{s},

e_{s} = ⋁_{c∈C} c.  (4)

and J_{s} = J_{e_{s}}, 2. e_{s^{∗}} = e_{s} for all s ∈ S,

3In [Sta15], we define condition (H) for another class of semigroups, namely the right LCM semigroups. Right LCM semigroups and inverse semigroups are related, but the intersection of their classes is empty (because right LCM semigroups are left cancellative and we assume that our inverse semigroups have a zero element). We note that a right LCM semigroup P satisfies condition (H) in the sense of [Sta15] if and only if its left inverse hull I_{l}(P) satisfies condition (H) in the sense of the above.

Documenta Mathematica 21 (2016) 809–840

3. e_{st} ⩽ ss^{∗}, t^{∗}t for all s, t ∈ S, and

4. e_{s^{∗}t}e_{t^{∗}r} ⩽ e_{s^{∗}r} for all s, t, r ∈ S.

Proof. To show the first statement, we need to show that any two covers give the same join. If J_{s} = {0}, there is nothing to do. So suppose that 0 ≠ e ∈ J_{s}, suppose that C is a cover for J_{s}, and let e_{C} = ⋁_{c∈C} c. Indeed, the element ee_{C}^{⊥} must be in J_{s}, and since it is orthogonal to all elements of C and C is a cover, ee_{C}^{⊥} must be 0. Hence we have

e = ee_{C} ∨ ee_{C}^{⊥} = ee_{C}

and so e ⩽ e_{C}. Now if K is another cover for J_{s} with join e_{K} and k ∈ K, we must have that k ⩽ e_{C}, and so e_{K} ⩽ e_{C}. Since the argument is symmetric, we have proven the first statement. To prove the second statement, if e ∈ J_{s} then we have

ses^{∗} = es^{∗} = (se)^{∗} = e

and so

s^{∗}e = s^{∗}(ses^{∗}) = es^{∗}ss^{∗} = es^{∗} = (se)^{∗} = e

and again by symmetry we have J_{s} = J_{s^{∗}} and so e_{s} = e_{s^{∗}}. To prove the third statement, we notice

ss^{∗}e_{st} = ss^{∗}ste_{st} = ste_{st} = e_{st}
e_{st}t^{∗}t = ste_{st}t^{∗}t = stt^{∗}te_{st} = ste_{st} = e_{st}.

For the fourth statement, we calculate (using 2)

e_{s^{∗}t}e_{t^{∗}r} = s^{∗}te_{s^{∗}t}e_{t^{∗}r} = s^{∗}tt^{∗}re_{s^{∗}t}e_{t^{∗}r}
= s^{∗}tt^{∗}rr^{∗}te_{s^{∗}t}e_{t^{∗}r} = s^{∗}rr^{∗}te_{s^{∗}t}e_{t^{∗}r}
= s^{∗}re_{s^{∗}t}e_{t^{∗}r}

hence e_{s^{∗}t}e_{t^{∗}r} ⩽ s^{∗}r and so e_{s^{∗}t}e_{t^{∗}r} ⩽ e_{s^{∗}r}.

In what will be a crucial step to obtaining a trace from an invariant mean, we now obtain a relationship between e_{st} and e_{ts}.

Lemma 4.3. Let S be Boolean inverse monoid which satisfies condition (H). Then for all s, t ∈ S, we have that s^{∗}e_{st}s = e_{ts}.

Proof. Suppose that e ∈ J_{ts}. Then tse = e, and so

(st)ses^{∗} = ses^{∗}

hence ses^{∗} ∈ J_{st}. If C is a cover of J_{st} and f ∈ J_{ts}, there must exist c ∈ C such that c(sfs^{∗}) ≠ 0. Hence

css^{∗}sfs^{∗} ≠ 0
ss^{∗}csfs^{∗} ≠ 0
s^{∗}csf ≠ 0

and so we see that s^{∗}Cs is a cover for J_{ts}. By Lemma 4.2,

e_{ts} = ⋁_{c∈C} s^{∗}cs = s^{∗} ( ⋁_{c∈C} c) s = s^{∗}e_{st}s.

Lemma 4.3 and Lemma 4.2.3 imply that for all s, t ∈ S and all μ ∈ M(S), we have μ(e_{st}) = μ(e_{ts}).

Remark 4.4. We are thankful to Ganna Kudryavtseva for pointing out to us that the proofs Lemmas 4.2 and 4.3 can be simplified by using the fact from [KL14, Theorem 8.20] that a Boolean inverse monoid S satisfies condition (H) if and only if every pair of elements in S has a meet (see also [Ste10, Proposition 3.7] for another wording of this fact). From this, one can see that for all s ∈ S we have

e_{s} = s ∧ (s^{∗}s) = s ∧ (ss^{∗}).

Definition 4.5. Let A be a C*-algebra. A bounded linear functional τ : A → ℂ is called a trace if

1. τ(a^{∗}a) ≥ 0 for all a ∈ A,

2. τ(ab) = τ(ba) for all a, b ∈ A.

A trace τ is said to be faithful if τ(a^{∗}a) > 0 for all a ≠ 0. A trace τ on a unital C*-algebra is called a tracial state if τ(1) = 1. The set of all tracial states of a C*-algebra A is denoted T (A).

We are now able to define a trace on C_{B}^{∗}(S) for each μ ∈ M(S).

Proposition 4.6. Let S be Boolean inverse monoid which satisfies condition (H), and let μ ∈ M(S). Then there is a trace τ_{μ} on C_{B}^{∗}(S) such that

τ_{μ}(δ_{s}) = μ(e_{s}) for all s ∈ S.

If μ is faithful, then the restriction of τ_{μ} to C_{red}^{∗}(G_{tight}(S)) is a faithful trace.

Proof. We define τ_{μ} to be as above on the generators δ_{s} of C_{B}^{∗}(S), and extend it to B :=span{δ_{s} | s ∈ S}, a dense ∗-subalgebra of C_{B}^{∗}(S).

We first show that τ_{μ}(δ_{s}δ_{t}) = τ_{μ}(δ_{t}δ_{s}). Indeed, by Lemmas 4.2 and 4.3, we have

τ_{μ}(δ_{s}δ_{t}) = μ(e_{st}) = μ(e_{st}ss^{∗}) = μ(e_{st}ss^{∗}e_{st}) = μ((e_{st}s)(e_{st}s)^{∗})
= μ((e_{st}s)^{∗}(e_{st}s)) = μ(s^{∗}e_{st}s) = μ(e_{ts}) = τ_{μ}(δ_{t}δ_{s}).

Since τ_{μ} is extended linearly to B, we have that τ_{μ}(ab) = τ_{μ}(ba) for all a, b ∈ B. Let F be a finite index set and take x = ∑_{i∈F} a_{i}δ_{s_{i}} in B. We will show that τ_{μ}(x^{∗}x) ≥ 0. For i, j ∈ F, we let e_{ij} = e_{s_{i}^{∗}s_{j}} and note that e_{ij} = e_{ji}. We calculate:

x^{∗}x = (∑_{s∈S} \overline{a_{i}}δ_{s_{i}^{∗}}) (∑_{j∈F} a_{j}δ_{s_{j}} )

= ∑_{i,j∈F} \overline{a_{i}}a_{j}δ_{s_{i}^{∗}s_{j}}
τ_{μ}(x^{∗}x) = ∑_{i,j∈F} \overline{a_{i}}a_{j}μ(e_{ij})
= ∑_{i∈F} |a_{i}|^{2}μ(e_{ii}) + ∑_{i,j∈F,i≠j}(\overline{a_{i}}a_{j} + \overline{a_{j}}a_{i})μ(e_{ij}).

We will show that this sum is positive by using an orthogonal decomposition of the e_{ij}. Let F_{≠}^{2} = {{i, j} ⊂ F | i ≠ j}, and let D(F_{≠}^{2}) = {(A, B) | A ∪ B = F_{≠}^{2}, A ∩ B = ∅}. For a = {i, j} ∈ F_{≠}^{2}, let e_{a} = e_{ij}. We have

e_{ij} = e_{ij} ⋁_{(A,B)∈D(F_{≠}^{2})} ( ∏_{a∈A,b∈B} e_{a}e_{b}^{⊥})

where the join is an orthogonal join. Of course, the above is only nonzero when {i, j} ∈ A. We also notice that

e_{ii} ⩾ ⋁_{\begin{matrix}(A,B)∈D(F_{≠}^{2}) \\ i∈∪A \end{matrix}} ( ∏_{a∈A,b∈B} e_{a}e_{b}^{⊥})

and so τ_{μ}(x^{∗}x) is larger than a linear combination of terms of the form μ (∏_{a∈A,b∈B} e_{a}e_{b}^{⊥}) for partitions (A, B) of F_{≠}^{2}: specifically, τ_{μ}(x^{∗}x) is greater than or equal to

∑_{(A,B)∈D(F_{≠}^{2})} [( ∑_{i∈∪A} |a_{i}|^{2} + ∑_{a=\{j,k\}∈A} (\overline{a_{i}}a_{j} + \overline{a_{j}}a_{i})) μ ( ∏_{a∈A,b∈B} e_{a}e_{b}^{⊥})] (5)

If a term ∏_{a∈A,b∈B} e_{a}e_{b}^{⊥} is not zero, then we claim that the relation

i ∼ j if and only if i = j or {i, j} ∈ A

is an equivalence relation on ∪A. Indeed, suppose that i, j, k ∈ ∪A are all pairwise nonequal and {i, j}, {j, k} ∈ A. By Lemma 4.2.4, e_{ij}e_{jk} ⩽ e_{ik} and since the product is nonzero, we must have that {i, k} ∈ A. Writing [∪A] for the set of equivalence classes, we have

∑_{i∈∪A} |a_{i}|^{2} + ∑_{a=\{j,k\}∈A} (\overline{a_{i}}a_{j} + \overline{a_{j}}a_{i}) = ∑_{C∈[∪A]} (∑_{i∈C} |a_{i}|^{2} + ∑_{\begin{matrix}i,j∈C \\ i≠j \end{matrix}} (\overline{a_{i}}a_{j} + \overline{a_{j}}a_{i}))

= ∑_{C∈[∪A]} |∑_{i∈C} a_{i}|^{2} .

Hence, τ_{μ}(x^{∗}x) ≥ 0, and τ_{μ} is positive on B. Hence, τ_{μ} extends to a trace on C_{B}^{∗}(S). The above calculation shows that if μ is faithful, then τ_{μ} is faithful on B. A short calculation shows that E(δ_{s}) = δ_{e_{s}}, where E is as in (1). Furthermore, it is clear that on B we have that τ_{μ} = τ_{μ} ∘ E, and so we will show that τ_{μ} is faithful on C_{red}^{∗}(G_{tight}(S)) if we show that τ_{μ}(a) > 0 for all nonzero positive a ∈ C(Ê_{tight}(S)). If a ∈ C(Ê_{tight}(S)) is positive, then it is bounded above zero on some clopen set given by D_{e} for some e ∈ E(S). Hence, τ_{μ}(a) ≥ τ_{μ}(δ_{e}) = μ(e) which must be strictly positive because μ is faithful.

We now show that given a trace on C_{B}^{∗}(S) we can construct an invariant mean on S.

Proposition 4.7. Let S be Boolean inverse monoid, let π_{u} : S → C_{B}^{∗}(S) be the universal Boolean monoid representation of S, and take τ ∈ T (C_{B}^{∗}(S)). Then the map μ_{τ} : E(S) → [0, ∞) defined by

μ_{τ} (e) = τ(π_{u}(e)) = τ(δ_{e})

is a normalized invariant mean on S. If τ is faithful then so is μ_{τ}.

Proof. That μ_{τ} takes positive values follows from τ being positive. We have

μ_{τ} (s^{∗}s) = τ(π_{u}(s^{∗}s)) = τ(π_{u}(s^{∗})π_{u}(s))
= τ(π_{u}(s)π_{u}(s^{∗})) = τ(π_{u}(ss^{∗}))
= μ_{τ} (ss^{∗}).

Also, if e, f ∈ E(S) with ef = 0, then

μ_{τ} (e ∨ f) = τ(π_{u}(e ∨ f)) = τ(π_{u}(e) + π_{u}(f))
= τ(π_{u}(e)) + τ(π_{u}(f))
= μ_{τ} (e) + μ_{τ} (f).

If τ is faithful and e ≠ 0, τ(δ_{e}) > 0 because δ_{e} is positive and nonzero, and so μ_{τ} is faithful.

Proposition 4.8. Let S be Boolean inverse monoid which satisfies condition (H). Then the map

μ ↦ τ_{μ} ↦ μ_{τ_{μ}}

is the identity on M(S).

Proof. This is immediate, since if μ ∈ M(S) and e ∈ E(S) we have

μ_{τ_{μ}}(e) = τ_{μ}(π_{u}(e)) = τ_{μ}(δ_{e}) = μ(e).

Given the above, one might wonder under which circumstances we have that T (C_{B}^{∗}(S)) ≅ M(S). This is not true in the general situation – take for example S to be the group ℤ_{2} = {1, −1} with a zero element adjoined – this is a Boolean inverse monoid. Here M(S) consists of one element, namely the function which takes the value 1 on 1 and the value 0 on the zero element. The C*-algebra of S is the group C*-algebra of ℤ_{2}, which is isomorphic to ℂ^{2}, a C*-algebra with many traces (taking the dot product of an element of ℂ^{2} with any nonnegative vector whose entries add to 1 determines a normalized trace on ℂ^{2}). One can still obtain this isomorphism using the following.

Definition 4.9. Let G be an étale groupoid. A regular Borel probability measure ν on G^{(0)} is called G-invariant if for every bisection U one has that ν(r(U)) = ν(d(U)). The affine space of all regular G-invariant Borel probability measures is denoted IM(G).

The following is a special case of [KR06, Proposition 3.2].

Theorem 4.10. (cf [KR06, Proposition 3.2]) Let G be a Hausdorff principal étale groupoid with compact unit space. Then

T (C_{red}^{∗}(G)) ≅ IM(G)

For τ ∈ T (C_{red}^{∗}(G)) the image of τ under the above isomorphism is the regular Borel probability measure ν whose existence is guaranteed by the Riesz representation theorem applied to the positive linear functional on C(G^{(0)}) given by restricting τ.

For a proof of Theorem 4.10 in the above form, see [Put, Theorem 3.4.5]. For us, the groupoid G_{tight}(S) satisfies all of the conditions in Theorem 4.10, except possibly for being principal. Also note that in the general case, C_{red}^{∗}(G_{tight}) may not be isomorphic to C_{B}^{∗}(S). So if we restrict our attention to Boolean inverse monoids which have principal tight groupoids and for which C_{red}^{∗}(G_{tight}(S)) ≅ C_{B}^{∗}(S) (that is to say, Boolean inverse monoids for which G_{tight}(S) satisfies weak containment), we can obtain the desired isomorphism. While this may seem like a restrictive set of assumptions, they are all satisfied for the examples we consider here.

Proposition 4.11. Let S be Boolean inverse monoid which satisfies condition (H), and suppose ν ∈ IM(G_{tight}(S)). Then the map η_{ν} : E(S) → [0, ∞) defined by

η_{ν}(e) = ν(D_{e}^{θ})

is a normalized invariant mean on S. The map that sends ν ↦ η_{ν} is an affine isomorphism of IM(G_{tight}(S)) and M(S).

Proof. That η_{ν}(s^{∗}s) = η_{ν}(ss^{∗}) follows from invariance of ν applied to the bisection Θ(s, D_{s^{∗}s}), and that η_{ν} is additive over orthogonal joins follows from the fact that ν is a measure. This map is clearly affine. Suppose that η_{ν} = η_{κ} for ν, κ ∈ IM(G_{tight}(S)). Then ν, κ agree on all sets of the form D_{e}^{θ}, and since these sets generate the topology on Ê_{tight}(S), ν and κ agree on all open sets. Since they are regular Borel probability measures they must be equal, and so ν ↦ η_{ν} is injective. To get surjectivity, let μ be an invariant mean, and let τ_{μ} be as in Proposition 4.6. Then restricting τ_{μ} to C(Ê_{tight}(S)) and invoking the Riesz representation theorem gives us a regular invariant probability measure ν on Ê_{tight}(S), and we must have η_{ν} = μ.

Corollary 4.12. Let G be an ample Hausdorff groupoid. Then IM(G) ≅ M(G^{a}).

So the invariant means on the ample semigroup of an ample Hausdorff groupoid are in one-to-one correspondence with the G-invariant measures.

Theorem 4.13. Let S be Boolean inverse monoid which satisfies condition (H). Suppose that G_{tight}(S) is principal, and that C_{red}^{∗}(G_{tight}(S)) ≅ C_{B}^{∗}(S). Then

T (C_{B}^{∗}(S)) ≅ M(S)

via the map which sends τ to μ_{τ} as in Proposition 4.7. In addition, both are isomorphic to IM(G_{tight}(S)).

Proof. This follows from Theorem 4.10 and Proposition 4.11.

There are many results in the literature concerning traces which now apply to our situation.

Corollary 4.14. Let S be Boolean inverse monoid which satisfies condition (H). If S admits a faithful invariant mean, then C_{red}^{∗}(G_{tight}(S)) is stably finite. If in addition G_{tight}(S) satisfies weak containment, C_{B}^{∗}(S) is stably finite.

Proof. If μ is a faithful invariant mean, then after normalizing one obtains a faithful trace on C_{red}^{∗}(G_{tight}(S)) by Proposition 4.6. Now the result is standard, see for example [LLR00, Exercise 5.2].

Corollary 4.15. Let S be Boolean inverse monoid which satisfies condition (H). If C_{B}^{∗}(S) is stably finite and exact, then S has an invariant mean.

Proof. This is a consequence of the celebrated result of Haagerup [Haa91] when applied to Proposition 4.7

For the undefined terms above, we direct the interested reader to [BO08]. We also note that exactness of C_{B}^{∗}(S) has recently been considered in [Li16] and [AD16a].

5 Examples

5.1 AF inverse monoids

This is a class of Boolean inverse monoids introduced in [LS14] motivated by the construction of AF C*-algebras from Bratteli diagrams. A Bratteli diagram is an infinite directed graph B = (V, E, r, s) such that

1. V can be written as a disjoint union of finite sets V = ∪_{n≥0}V_{n}

2. V_{0} consists of one element v_{0}, called the root,

3. for all edges e ∈ E, s(e) ∈ V_{i} implies that r(e) ∈ V_{i+1} for all i ≥ 0, and

4. for all i ≥ 1 and all v ∈ V_{i}, both r^{−1}(v) and s^{−1}(v) are finite and nonempty.

We also denote s^{−1}(V_{i}) := E_{i}, so that E = ∪_{n≥0}E_{n}. Let E^{∗} be the set of all finite paths in B, including the vertices (treated as paths of length zero). For v, w ∈ V ∪ E, let vE^{∗} denote all the paths starting with v, let E^{∗}w be all the paths ending with w, and let vE^{∗}w be all the paths starting with v and ending with w. Given a Bratteli diagram B = (V, E, r, s) we construct a C*-algebra as follows. We let

A_{0} = ℂ

A_{1} = ⨁_{v∈V_{1}} 𝕄_{|r^{−1}(v)|},

and define k_{1}(v) = |r^{−1}(v)| for all v ∈ V_{1}. For an integer i > 1 and v ∈ V_{i}, let

k_{i}(v) = ∑_{γ∈r^{−1}(v)} k_{i−1}(s(γ)).  (6)

Define

A_{i} = ⨁_{v∈V_{i}} 𝕄_{k_{i}(v)}

Now for all i ≥ 0, one can embed A_{i} ↪ A_{i+1} by viewing, for each v ∈ V_{i+1}

⨁_{γ∈r^{−1}(v)} 𝕄_{k_{i}(s(γ))} ⊂ 𝕄_{k_{i+1}(v)}

where the algebras in the direct sum are orthogonal summands along the diagonal in 𝕄_{k_{i+1}(v)}. So A_{0} ↪ A_{1} ↪ A_{1} ↪ ⋯ can be viewed as an increasing union of finite dimensional C*-algebra, all of which can be realized as subalgebras of B(H) for the same H, and so we can form the norm closure of the union

A_{B} := \overline{⋃_{n≥0} A_{n}}.

This C*-algebra is what is known as an AF algebra, and every unital AF algebra arises this way from some Bratteli diagram. The AF algebra A_{B} can always be described as the C*-algebra of a principal groupoid derived from B, see [Ren80] and [ER06]. We reproduce this construction here. Let X_{B} denote the set of all infinite paths in B which start at the root. When given the product topology from the discrete topologies on the E_{n}, this is a compact Hausdorff totally disconnected space. For α ∈ v_{0}E^{∗}, we let C(α) = {x ∈ X_{B} | x_{i} = α_{i} for all i = 0, …, |α| − 1}. Sets of this form are clopen and form a basis for the topology on X_{B}. For n ∈ ℕ, let

R_{B}^{(n)} = {(x, y) ∈ X × X | x_{i} = y_{i} for all i ≥ n + 1}

so a pair of infinite paths (x, y) is in R_{B}^{(n)} if and only if x and y agree after the vertices on level n. Clearly, R_{B}^{(n)} ⊂ R_{B}^{(n+1)}, and so we can form their union

R_{B} = ⋃_{n∈ℕ} R_{B}^{(n)}.

This is an equivalence relation, known as tail equivalence on X_{B}. For v ∈ V \ {v_{0}} and α, β ∈ v_{0}E^{∗}v, define

C(α, β) = {(x, y) ∈ R_{B} | x ∈ C(α), y ∈ C(β)}

sets of this type form a basis for a topology on R_{B}, and with this topology R_{B} is a principal Hausdorff étale groupoid with unit space identified with X_{B}, and

C^{∗}(R_{B}) ≅ C_{red}^{∗}(R_{B}) ≅ A_{B}.

In [LS14], a Boolean inverse monoid is constructed from a Bratteli diagram, mirroring the above construction. We will present this Boolean inverse monoid in a slightly different way which may be enlightening. Let B = (V, E, r, s) be a Bratteli diagram. Let S_{0} be the Boolean inverse monoid (in fact, Boolean algebra) {0, 1}. For each i ≥ 1, let

S_{i} = ⨁_{v∈V_{i}} I(v_{0}E^{∗}v)

where as in Section 2.1, I(X) denotes the set of partially defined bijections on X.

If v ∈ V_{i+1} and γ ∈ r^{−1}(v) then one can view I(v_{0}E^{∗}γ) as a subset of I(v_{0}E^{∗}v), and if η ∈ r^{−1}(v) with γ ≠ η, I(v_{0}E^{∗}γ) and I(v_{0}E^{∗}η) are orthogonal. Furthermore, I(v_{0}E^{∗}γ) can be identified with I(v_{0}E^{∗}s(γ)) Hence the direct sum over r^{−1}(v) can be embedded into I(v_{0}E^{∗}v):

⨁_{γ∈r^{−1}(v)} I(v_{0}E^{∗}s(γ)) ↪ I(v_{0}E^{∗}v).  (7)

This allows us to embed S_{i} ↪ S_{i+1}

⨁_{v∈V_{i}} I(v_{0}E^{∗}v) ↪ ⨁_{w∈V_{i+1}} I(v_{0}E^{∗}w)

where an element ϕ in a summand I(v_{0}E^{∗}v) gets sent to |s^{−1}(v)| summands on the right, one for each γ ∈ s^{−1}(v): ϕ will be sent to the summand inside I(v_{0}E^{∗}s(γ)) corresponding to v in left hand side of the embedding from (7). We then define

I(B) = lim_{→}(S_{i} ↪ S_{i+1})

This is a Boolean inverse monoid [LS14, Lemma 3.13]. As a set I(B) is the union of all the S_{i}, viewed as an increasing union via the identifications above. In [LS14, Remark 6.5], it is stated that the groupoid one obtains from I(B) (i.e., G_{tight}(I(B))) is exactly tail equivalence. We provide the details of that informal discussion here. We will describe the ultrafilters in E(I(B)), a Boolean algebra. For v ∈ V_{i} and a path α ∈ v_{0}E^{∗}v, let e_{α} = Id_{{α}} ∈ I(v_{0}E^{∗}v). As v ranges over all of V_{i} and α ranges over all of v_{0}E^{∗}v, these idempotents form a orthogonal decomposition of the identity of I(B). Hence, given an ultrafilter ξ and i > 0 there exists one and only one path, say α_{ξ}^{(i)} ending at level i with e_{α_{ξ}^{(i)}} ∈ ξ. Furthermore,

if j > i, we must have that α_{ξ}^{(i)} is a prefix of α_{ξ}^{(j)}, because products in an ultrafilter cannot be zero. So for x ∈ X_{B}, if we define

ξ_{x} = {e_{α} | α is a prefix of x}

then we have that

Ê_{∞}(I(B)) = {ξ_{x} | x ∈ X_{B}}

By [EP16, Proposition 2.6], the set

{U({e_{α}}, ∅) | α is a prefix of x}

is a neighbourhood basis for ξ_{x}. The map λ : X_{B} → Ê_{∞}(I(B)) given by λ(x) = ξ_{x} is a bijection, and since U({e_{α}}, ∅) = λ(C(α)), it is a homeomorphism. If ϕ ∈ S_{i} such that ϕ^{∗}ϕ ∈ ξ_{x}, then we must have that one component of ϕ is in I(v_{0}E^{∗}r(x_{i})), and we must have that

θ_{ϕ}(ξ_{x}) = ξ_{ϕ(x_{0}x_{1}...x_{i})x_{i+1}x_{i+2}...}  (8)

Finally, we claim that R_{B} is isomorphic to G_{tight}(I(B)). We define a map

Φ : G_{tight}(I(B)) → R_{B}
Φ([ϕ, ξ_{x}]) ↦ (ϕ(x_{0}x_{1} … x_{i})x_{i+1}x_{i+2} …, x)

where ϕ and x are as in (8). If Φ([ϕ, ξ_{x}]) = Φ([ψ, ξ_{y}]), then clearly we must have ξ_{x} = ξ_{y}. We must also have that ϕ, ψ ∈ S_{i}, and ϕe_{x_{0}x_{1}...x_{i}} = ψe_{x_{0}x_{1}...x_{i}}, hence [ϕ, ξ_{x}] = [ψ, ξ_{y}]. It is straightforward to verify that Φ is surjective and bicontinuous, and so R_{B} ≅ G_{tight}(I(B)). Since they are both étale, their C*- algebras must be isomorphic. Hence with the above discussion, we have proven the following.

Theorem 5.1. Let B be a Bratteli diagram. Then

C_{B}^{∗}(I(B)) ≅ A_{B}.

Furthermore, every unital AF algebra is isomorphic to the universal C*-algebra of a Boolean inverse monoid of the form I(B) for some B.

Recall that a compact convex metrizable subset X of a locally convex space is a Choquet simplex if and only if for each x ∈ X there exists a unique measure ν concentrated on the extreme points of X for which x is the center of gravity of X for ν [Phe01]. Now we can use the following seminal result of Blackadar to make a statement about the set of normalized invariant means for AF inverse monoids.

Theorem 5.2. (Blackadar, see [Bla80, Theorem 3.10]) Let Δ be any metrizable Choquet simplex. Then there exists a unital simple AF algebra A such that T (A) is affinely isomorphic to Δ.

Corollary 5.3. Let Δ be any metrizable Choquet simplex. Then there exists an AF inverse monoid S such that M(S) is affinely isomorphic to Δ.

Proof. This result follows from Theorem 4.13 because G_{tight}(S) is Hausdorff, amenable, and principal for every AF inverse monoid S.

5.2 The 3 × 3 matrices

This example is a subexample of the previous example, but it will illustrate how we approach the following two examples. Let I_{3} denote the symmetric inverse monoid on the three element set {1, 2, 3}. This is a Boolean inverse monoid which satisfies condition (H), and we define a map π : I_{3} → 𝕄_{3} by saying that

π(ϕ)_{ij} = \begin{cases}1 & if ϕ(j) = i \\ 0 & otherwise. \end{cases}

Then it is straightforward to verify that π is in fact the universal Boolean inverse monoid representation of I_{3}. Now instead consider the subset R_{3} ⊂ I_{3} consisting of the identity, the empty function, and all functions with domain consisting of one element. Then R_{3} is an inverse monoid, and π(R_{3}) is the set of all matrix units together with the identity matrix and zero matrix. When restricted to R_{3}, π is the universal tight representation of R_{3}. Hence C_{tight}^{∗}(R_{3}) ≅ C_{B}^{∗}(I_{3}) ≅ 𝕄_{3}. There is only one invariant mean μ on I_{3} – for an idempotent Id_{U} ∈ I_{3}, we have μ(Id_{U}) = \frac{1}{3}|U|. The tight groupoid of R_{3} is the equivalence relation {1, 2, 3} × {1, 2, 3}, which is principal – we also have that G_{tight}(R_{3})^{a} ≅ I_{3}. The unique invariant mean on I_{3} is identified with the unique normalized trace on 𝕄_{3}. Our last two examples follow this mold, where we have an inverse monoid S which generates a C*-algebra C_{tight}^{∗}(S), and we relate the traces of C_{tight}^{∗}(S) to the invariant means of G_{tight}(S)^{a}.

5.3 Self-similar groups

Let X be a finite set, let G be a group, and let X^{∗} denote the set of all words in elements of X, including an empty word ∅. Let X^{ω} denote the Cantor set of one-sided infinite words in X, with the product topology of the discrete topology on X. For α ∈ X^{∗}, let C(α) = {αx | x ∈ X^{ω}} – sets of this type are called cylinder sets and form a clopen basis for the topology on X. Suppose that we have a faithful length-preserving action of G on X^{∗}, with (g, α) ↦ g · α, such that for all g ∈ G, x ∈ X there exists a unique element of G, denoted g|_{x}, such that for all α ∈ X^{∗}

g(xα) = (g · x)(g|_{x} · α).

In this case, the pair (G, X) is called a self-similar group. The map G×X → G, (g, x) ↦ g|_{x} is called the restriction and extends to G × X^{∗} via the formula

g|_{α_{1}···α_{n}} = g|_{α_{1}} |_{α_{2}} ⋯ |_{α_{n}}

and this restriction has the property that for α, β ∈ X^{∗}, we have

g(αβ) = (g · α)(g|_{α} · β).

The action of G on X^{∗} extends to an action of G on X^{ω} given by

g · (x_{1}x_{2}x_{3} …) = (g · x_{1})(g|_{x_{1}} · x_{2})(g|_{x_{1}x_{2}} · x_{3}) ⋯

In [Nek09], Nekrashevych associates a C*-algebra to (G, X), denoted O_{G,X}, which is the universal C*-algebra generated by a set of isometries {s_{x}}_{x∈X} and a unitary representation {u_{g}}_{g∈G} satisfying

(i) s_{x}^{∗}s_{y} = 0 if x ≠ y, (ii) ∑_{x∈X} s_{x}s_{x}^{∗} = 1,

(iii) u_{g}s_{x} = s_{g·x}u_{g|_{x}}.

One can also express O_{G,X} as the tight C*-algebra of an inverse semigroup. Let

S_{G,X} = {(α, g, β) | α, β ∈ X^{∗}, g ∈ G} ∪ {0}.

This set becomes an inverse semigroup when given the operation

(α, g, β)(γ, h, ν) = \begin{cases}(α(g · γ′), g|_{γ′} h, ν), & if γ = βγ′, \\ (α, g(h^{−1}|_{β′})^{−1}, ν(h^{−1} · β′)), & if β = γβ′, \\ 0 & otherwise \end{cases}

with

(α, g, β)^{∗} = (β, g^{−1}, α).

Here, E(S_{X,G}) = {(α, 1_{G}, α) | α ∈ X^{∗}}, and the tight spectrum Ê_{tight}(S_{G,X}) is homeomorphic X^{ω} by the identification

x ∈ X^{ω} ↦ {(α, 1_{G}, α) ∈ E(S_{G,X}) | α is a prefix of x} ∈ Ê_{tight}(S_{G,X}).

If θ is the standard action of S_{G,X} on Ê_{tight}(S_{G,X}), then D_{(α,1_{G},α)}^{θ} = C(α). If s = (α, g, β) ∈ S_{X,G}, then

θ_{s} : C(β) → C(α)
θ_{s}(βx) = α(g · x)

It is shown in [EP14] that O_{G,X} is isomorphic to C_{tight}^{∗}(S_{G,X}). We show that the universal tight representation of S_{G,X} is faithful. This will be accomplished if we can show that the map from S_{G,X} to G_{tight}(S_{G,X})^{a} given by

s ↦ Θ(s, D_{s^{∗}s}^{θ})

is injective. If s = (α, g, β), then

Θ(s, D_{s^{∗}s}^{θ}) = {[(α, g, β), βx] | x ∈ X^{ω}}.

It is straightforward that d(Θ(s, D_{s^{∗}s}^{θ})) = C(β) and r(Θ(s, D_{s^{∗}s}^{θ})) = C(α). Suppose we have another element t = (γ, h, η) such that Θ(s, D_{s^{∗}s}^{θ}) = Θ(t, D_{t^{∗}t}^{θ}). Since these two bisections are equal, their sources (resp. ranges) must be equal, so C(β) = C(η) (resp. C(α) = C(γ)). Hence, α = γ and β = η. Since r and d are both bijective on these slices, we must have that for all βx ∈ C(β), α(g · x) = α(h · x). Hence for all x ∈ X^{ω}, we must have that g · x = h · x. The action of G on X^{∗} is faithful, so the induced action of G on X^{ω} is also faithful, hence g = h and so t = s. As it stands, the Boolean inverse monoid G_{tight}(S_{G,X})^{a} cannot have any invariant means. This is because the subalgebra of O_{G,X} generated by {s_{x} | x ∈ X} is isomorphic to the Cuntz algebra O_{|X|}, and a trace on O_{G,X} would have to restrict to a trace on O_{|X|}, which is purely infinite and hence has no traces. To justify the inclusion of this example in this paper about invariant means, we restrict to an inverse subsemigroup of S_{G,X} whose corresponding ample semigroup will admit an invariant mean. Let

S_{G,X}^{=} = {(α, g, β) ∈ S_{G,X} | |α| = |β|} ∪ {0}.

One can easily verify that this is closed under product and involution, and so is a inverse subsemigroup of S_{G,X}, with the same set of idempotents as S_{G,X}. If α, β ∈ X^{∗}, |α| = |β|, and g ∈ G, then

(α, g, β)^{∗}(α, g, β) = (β, 1_{G}, β), (α, g, β)(α, g, β)^{∗} = (α, 1_{G}, α).

If μ were an invariant mean on G_{tight}(S_{G,X}^{=})^{a}, then we would have to have, for all α, β ∈ X^{∗} and |α| = |β|, that μ(C(α)) = μ(C(β)). Moreover, for a given length n, the set {C(α) | |α| = n} forms a disjoint partition of X^{ω}, and so we must have

μ(C(α)) = |X|^{−|α|}.  (9)

Any clopen subset of X^{ω} must be a finite disjoint union of cylinders. Hence the map μ on E(G_{tight}(S_{G,X}^{=})^{a}) determined by (9) is an invariant mean, and is in fact the unique invariant mean on G_{tight}(S_{G,X}^{=})^{a}. In the general case, it is possible for G_{tight}(S_{G,X}^{=}) to be neither Hausdorff nor principal. We now give an explicit example where we get a unique trace to go along with our unique invariant mean.

Example 5.4. (The 2-odometer) Let X = {0, 1} and let ℤ = ⟨z⟩ be the group of integers with identity e written multiplicatively. The 2-odometer is the self-similar group (ℤ, X) determined by

z · 0 = 1 z|_{0} = e
z · 1 = 0 z|_{1} = z.

If one views a word α ∈ X^{∗} as a binary number (written backwards), then z · α is the same as 1 added to the binary number for α, truncated to the length of α if needed. If such truncation is not needed, z|_{α} = e, but if truncation is needed, z|_{α} = z. The action of ℤ on {0, 1}^{ω} induced by the 2-odometer is the familiar Cantor minimal system of the same name. For x ∈ {0, 1}^{ω} we have

z · x = \begin{cases}000 ⋯ & if x_{i} = 1 for all i \\ 00 ⋯ 01x_{i+1}x_{i+2} ⋯ & if x_{i} = 0 and x_{j} = 1 for all j < i \end{cases}

This action of ℤ is free (i.e. z^{n} · x = x implies n = 0) and minimal (i.e. the set {z^{n} · x | n ∈ ℤ} is dense in {0, 1}^{ω} for all x ∈ {0, 1}^{ω}).

Lemma 5.5. The groupoid of germs G_{tight}(S_{ℤ,X}^{=}) is principal.

Proof. Take x, y ∈ {0, 1}^{ω} and suppose that we have α, β ∈ {0, 1}^{∗} with |α| = |β| and n ∈ ℤ such that [(α, z^{n}, β), x] ∈ G_{tight}(S_{ℤ,X}^{=}) and r([(α, z^{n}, β), x]) = y. This implies that x = βv for some v ∈ {0, 1}^{ω}, and that y = α(z^{n} · v). Suppose we can find another germ from x to y, that is, suppose we have γ, η ∈ {0, 1}^{∗} with |γ| = |η| and m ∈ ℤ such that [(γ, z^{n}, η), x] ∈ G_{tight}(S_{ℤ,X}) and r([(γ, z^{n}, η), x]) = y. Again we can conclude that x = ηu for some u ∈ {0, 1}^{ω}, and that y = γ(z^{m} · u). There are two cases. Suppose first that β = ηδ for some δ ∈ {0, 1}^{∗}. Then ηδv = x = ηu, and so δv = u. We also have α(z^{n} · v) = y = γ(z^{m} · u). Because |α| = |β| ≥ |η| = |γ|, this implies that there exists ν ∈ {0, 1}^{∗} with |ν| = |δ| and α = γν. Hence ν(z^{n} · v) = z^{m} · u = (z^{m} · δ) z^{m}|_{δ} · v, which gives us that ν = z^{m} · δ and z^{n} · v = z^{m}|_{δ} · v, and since the action on {0, 1}^{ω} is free we have z^{n} = z^{m}|_{δ}. So we have that x ∈ C(β) = D_{(β,e,β)}^{θ}, and we calculate

(γ, z^{n}, η)(β, e, β) = (γ(z^{m} · δ), z^{m}|_{δ} , β) = (γν, z^{n}, β) = (α, z^{n}, β)
= (α, z^{n}, β)(β, e, β)

where the first equality is by the definition of the product. Hence [(α, z^{n}, β), x] = [(γ, z^{n}, η), x]. The case where β is shorter than η is similar. Hence, G_{tight}(S_{ℤ,X}^{=}) is principal.

It is routine to check that S_{ℤ,X}^{=} satisfies condition (H) (in fact, it is E*-unitary, see [ES16, Example 3.4]). The groupoid G_{tight}(S_{ℤ,X}^{=}) is amenable, see [ADR00, Proposition 5.1.1] and [EP13, Corollary 10.18]. Hence Theorem 4.13 applies, and there is only one normalized trace on C_{tight}^{∗}(S_{ℤ,X}^{=}), the one arising from the invariant mean. As the observant reader is no doubt aware at this point, C_{tight}^{∗}(S_{ℤ,X}^{=}) is nothing more than the crossed product C({0, 1}^{ω}) ⋊ ℤ arising from the usual odometer action [Nek04, Theorem 7.2], which has a unique normalized trace due to the dynamical system ({0, 1}^{ω}, ℤ) having a unique invariant measure (given by (9)).

5.4 Aperiodic tilings

We close with another example where the traces on the relevant C*-algebras are known beforehand, and hence give us invariant means. A tile is a closed subset of ℝ^{d} homeomorphic to the closed unit ball. A partial tiling is a collection of tiles in ℝ^{d} with pairwise disjoint interiors, and the support of a partial tiling is the union of its tiles. A patch is a finite partial tiling, and a tiling is a partial tiling with support equal to ℝ^{d}. If P is a partial tiling and U ⊂ ℝ^{d}, then let P (U) be the partial tiling of all tiles in P which intersect U. A tiling T is called aperiodic if T + x ≠ T for all 0 ≠ x ∈ ℝ^{d}. Let T be a tiling. We form an inverse semigroup S_{T} from T as follows. For a patch P ⊂ T and tiles t_{1}, t_{2} ∈ P we call the triple (t_{1}, P, t_{2}) a doubly pointed patch. We put an equivalence relation on such triples, by saying that (t_{1}, P, t_{2}) ∼ (r_{1}, Q, r_{2}) if there exists a vector x ∈ ℝ^{d} such that (t_{1} + x, P + x, t_{2} + x) = (r_{1}, Q, r_{2}), and let [t_{1}, P, t_{2}] denote the equivalence class of such a triple – this is referred to a doubly pointed patch class. Let

S_{T} = {[t_{1}, P, t_{2}] | (t_{1}, P, t_{2}) is doubly pointed patch } ∪ {0}

be the set of all doubly pointed patch classes together with a zero element. If [t_{1}, P, t_{2}], [r_{1}, Q, r_{2}] are two elements of S_{T}, we let

[t_{1}, P, t_{2}][r_{1}, Q, r_{2}] = \begin{cases}[t_{1}, P ∪ Q′, r′_{2}] & if there exists (r′_{1}, Q′, r′_{2}) ∈ [r_{1}, Q, r_{2}] \\ & such that r′_{1} = t_{2} and P ∪ Q′ is \\ & a patch in T + x for some x ∈ ℝ^{d} \\ 0 & otherwise, \end{cases}

and define all products involving 0 to be 0. Also, let [t_{1}, P, t_{2}]^{∗} = [t_{2}, P, t_{1}]. With these operations, S_{T} is an inverse semigroup. This inverse semigroup was defined by Kellendonk [Kel97a] [Kel97b], and is E*-unitary. Suppose there exists a finite set P of tiles each of which contain the origin in the interior such that for all t ∈ T, there exists x_{t} ∈ ℝ^{d} and p ∈ P such that t = p + x_{t}. In this case, P is called a set of prototiles for T. By possibly adding labels, we may assume that x_{t} and p are unique – we call x_{t} the puncture of t. Consider the set

X_{T} = {T − x_{t} | t ∈ T }

and put a metric on X_{T} by setting

d(T_{1}, T_{2}) = inf{1, ϵ | T_{1}(B_{1/ϵ}(0)) = T_{1}(B_{1/ϵ}(0))}

and let Ω_{punc} denote the completion of X_{T} in this metric (above, B_{r}(x) denotes the open ball in ℝ^{d} of radius r around x ∈ ℝ). One can show that all elements of Ω_{punc} are tilings consisting of translates of P which also contain an element of P and that the metric above extends to the same metric on Ω_{punc} – this is called the punctured hull of T. We make the following assumptions on T:

1. T has finite local complexity if for any r > 0, there are only finitely many patches in T with supports having outer radius less than r, up to translational equivalence.

2. T is repetitive if for every patch P ⊂ T, there exists R > 0 such that every ball of radius R in ℝ^{d} contains a translate of P.

3. T is strongly aperiodic if all elements of Ω_{punc} are aperiodic.

In this case Ω_{punc} is homeomorphic to the Cantor set. For a patch P ⊂ T and tile t ∈ P, let

U(P, t) = {T′ ∈ Ω_{punc} | P − x_{t} ⊂ T′}.

Figure 1: In the Robinson triangles version of the Penrose tiling, each triangle is always next to a similar triangle with which it forms a rhombus. Let P be the dark gray patch, and let P′ be the patch with the lighter gray tiles added. Then for any dark gray tile t, U(P, t) = U(P′, t)

Then these sets are clopen in Ω_{punc} and generate the topology. Let

R_{punc} = {(T_{1}, T_{1} + x) ∈ Ω_{punc} × Ω_{punc} | x ∈ ℝ^{d}}

and view this equivalence relation as a principal groupoid. Endow it with the topology inherited by viewing it as a subspace of Ω_{punc} × ℝ^{d}. For a patch P ⊂ T and t_{1}, t_{2} ∈ P, let

V (t_{1}, P, t_{2}) = {(T_{1}, T_{2}) ∈ R_{punc} | T_{1} ∈ U(P, t_{1}), T_{2} = T_{1} + x_{t_{1}} − x_{t_{2}}}

Then these sets are compact bisections in R_{punc}, and generate the topology on R_{punc}. This groupoid is Hausdorff, ample, and amenable [PS99]. The C*- algebra of R_{punc} was defined by Kellendonk in [Kel95] (denoted there A_{T}) and studied further in [KP00], [Put00], [Put10], [Phi05], [Sta14]. We proved in [EGS12, Theorem 3] that G_{tight}(S_{T}) ≅ R_{punc} – the universal tight representation of S_{T} maps [t_{1}, P, t_{2}] to the characteristic function of V (t_{1}, P, t_{2}).^{4} It is interesting to note that in this case that the universal tight representation may not be faithful. Suppose that we could find P ⊂ P′, both patches in T, and that P + x ⊂ T can only happen if P′ + x ⊂ T. Then for a tile t ∈ P, the two idempotents [t, P, t], [t, P′, t] are different elements in S_{T}, but are both mapped to the characteristic function of U(P, t) = U(P′, t) under the universal tight representation – indeed, [t, P′, t] is dense in [t, P, t], see Figure 1. We note that [Len08], [Exe09], and [LL13] address other cases where the tight representation may not be faithful. The C*-algebra A_{T} can be seen as the C*-algebra of a Boolean inverse monoid, namely G_{tight}(S_{T})^{a} – one could then rightly call this the Boolean inverse monoid associated to T. The traces of A_{T} are already well-studied, see [KP00], [Put00]. Often, as is the case with the Penrose tiling, there is a unique trace, see [Put00].

4The same result follows from [Len08, Section 9] combined with the fact that Lenz’s groupoid coincides with the tight groupoid when Ê_{tight}(S) = Ê_{∞}(S), see [LL13, Theorem 5.15].

Documenta Mathematica 21 (2016) 809–840

Theorem 5.6. Let T be a tiling which satisfies conditions 1–3 above, and let G_{tight}(S_{T})^{a} be the Boolean inverse monoid associated to T. Then M(G_{tight}(S_{T})^{a}) ≅ T (A_{T}) ≅ IM(R_{punc}).

Acknowledgment: I thank Thierry Giordano for an enlightening conversation about this work, and I also thank Ganna Kudryavtseva, Ruy Exel, and Daniel Lenz for comments and suggestions on a preprint version of this paper. Finally, I thank the referee for a thorough reading.

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Charles Starling University of Ottawa Department of Mathematics and Statistics 585 King Edward Ottawa, ON, Canada K1N 6N5

            cstar050@uottawa.ca