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On a groupoid construction for actions of certain

arXiv:funct-an/9305001v1 11 May 1993On a groupoid construction for actions of certaininverse semigroupsAlexandru NicaDepartment of MathematicsUniversity of CaliforniaBerkeley, California 94720(e-mail: nica@cory.berkeley.edu)AbstractWe consider a version of the notion of F-inverse semigroup (studied in the algebraictheory of inverse semigroups). We point out that an action of such an inverse semigroupon a locally compact space has associated a natural groupoid construction, very similarto the one of a transformation group.

We discuss examples related to Toeplitz algebrason subsemigroups of discrete groups, to Cuntz-Krieger algebras, and to crossed-productsby partial automorphisms in the sense of Exel.0

0IntroductionLet G be a discrete group, let P be a unital subsemigroup of G, and let W(G, P) denotethe Toeplitz (also called Wiener-Hopf) C*-algebra associated to (G, P); i.e., W(G, P) ⊆L(l2(P)) is the C*-algebra generated by the compression to l2(P) of the left regular repre-sentation Λ : l1(G) →L(l2(G)).A powerful tool for studying W(G, P) is a locally compact groupoid G introduced byP. Muhly and J. Renault in [7] (in a framework larger than the one of discrete groups),and which was shown in [7] to have C*red(G) ≃W(G, P).

We shall refer to G as to theWiener-Hopf groupoid associated to (G, P).On the other hand let us denote, for every x ∈G, by βx : {t ∈P | xt ∈P} →{s ∈P |x−1s ∈P} the partially defined left translation with x on P; and let us denote by SG,P thesemigroup of bijective transformations between subsets of P (with multiplication given bycomposition, defined where it makes sense) which is generated by (βx)x∈G. Then SG,P is aninverse semigroup, which encodes in some sense the action of G on P by left translations.The original motivation for this work was to understand the relation between theinverse semigroup SG,P and the Wiener-Hopf groupoid associated to (G, P).

It turns outthat:(A) there exists a remarkable action of SG,P on a compact space,and(B) there exists a very natural procedure of associating a locally compact groupoidto an action of an inverse semigroup in a class which contains SG,P,such that (A) and (B) together lead to the Wiener-Hopf groupoid.Moreover, there exists a natural unital ⋆-homomorphism Ψ from C*(SG,P) onto C*(G),where C*(SG,P ) denotes the enveloping C*-algebra of SG,P, and C*(G) the C*-algebra ofthe Wiener-Hopf groupoid. Verifying whether Ψ is faithful can be reduced to studying thesurjectivity of a map between compact spaces, and actually to comparing two subspacesof {0, 1}G (see Corollary 6.4 and Example 6.5 below).

This holds when G, viewed as aleft-ordered group with positive semigroup P, has some sort of lattice properties (as forinstance those considered in [9]), but is not true in general.Let us now return to the above (A) and (B). By an action of an inverse semigroup S ona locally compact space Ωwe shall understand a unital ⋆-homomorphism from S into the1

inverse semigroup of homeomorphisms between open subsets of Ω.Related to (A), let us assume for a moment that P does not contain a proper subgroupof G; then the action of SG,P mentioned at (A), let us call it Φ, is on a space Ωwhich isa compactification of P. Every α ∈SG,P can thus be viewed as a function between twosubsets Dom(α), Ran(α) ⊆P ⊆Ω, and we have the remarkable fact that Φ(α) is theunique continuous extension of α between the closures Dom(α) and Ran(α) in Ω.Related to (B): the fact which makes the groupoid construction associated to an actionof SG,P to be very simple, and indeed a straightforward generalization of the groupoid asso-ciated to a transformation group, is the following: every element of SG,P , except possibly thezero element, is majorized ( in the sense of the usual partial order on an inverse semigroup)by a unique maximal element. Inverse semigroups with this property have been studiedin the algebraic theory of inverse semigroups under the name of F-inverse semigroups (see[11], Section VII.6).There are various examples of F-inverse semigroups coming from the algebraic the-ory which can be considered (see Example 1.4 below).

Given the nature of the presentnote, it is probably even more interesting to remark that most of the singly generatedinverse semigroups have the property under consideration; moreover, the C*-algebra of thegroupoid associated to an action of a singly generated inverse semigroup is isomorphic tothe corresponding crossed-product C*-algebra by a partial automorphism, in the sense ofR. Exel [4].In [6], A. Kumjian has developed a method which associates a C*-algebra to a localiza-tion, i.e.

to an inverse semigroup S of homeomorphisms between open subsets of a locallycompact space, such that the family of domains (Dom(α))α∈S is a basis for the topologyof the space. The point of view of the present note is slightly different from the one of [6],and the examples we consider do not generally have the named localization property (seeSection 5 below).

For S in the common territory of the two approaches, the C*-algebra ofthe associated groupoid can be shown, however, to be isomorphic to the one constructed in[6] (see Theorem 5.1 below). This is for instance the case for the localizations giving theCuntz-Krieger C*-algebras OA.The paper is subdivided into sections as follows: after recalling in Section 1 some basicdefinitions, and giving some examples, the groupoid construction is presented in Section2.In Section 3 we discuss the example related to Toeplitz algebras, and in Section 4the one related to crossed products by partial automorphisms.Section 5 is devoted tothe relation with localizations; in 5.4 the example related to the Cuntz-Krieger algebras is2

discussed. Finally, in Section 6 we study the relation between C*(S) and the C*-algebraof the associated groupoid (and obtain in particular the facts about SG,P mentioned at thebottom of page 1).1Basic definitions and examplesA semigroup S is an inverse semigroup if for every α ∈S there exists a unique elementof S, denoted by α∗, such that αα∗α = α, α∗αα∗= α∗.

For the algebraic facts aboutinverse semigroups needed in this note, we shall use as a reference the monograph [11]. ForT a non-void set, we denote by IT the symmetric inverse semigroup on T, i.e.

the semigroupof all bijections between subsets of T (with multiplication given by composition, definedwhere it makes sense). This is a generic example, in the sense that every inverse semigroupcan be embedded into an IT .

(see [11], IV.1). On an arbitrary inverse semigroup we havea natural partial order, defined byα ≤βdef⇔β∗α = α∗α,α, β ∈S.

(1.1)α ≤β is equivalent to the fact that α is a restriction of β in every embedding of S into asymmetric inverse semigroup (see [11], II.1.6 and IV.1.10). An element α ∈S will be calledmaximal if there is no β ̸= α in S such that α ≤β.We shall consider only unital inverse semigroups, and the unit will be usually denotedby ǫ.

Also, an inverse semigroup may or may not have a zero element, which, if existing,is unique and will be denoted by θ ( αθ = θα = θ for all α ∈S). If α is an element ofthe inverse semigroup S, the expression “α is not zero element for S” will be used to meanthat, if S happens to have a zero element θ, then α ̸= θ.A unital inverse semigroup S is called an F-inverse semigroup if every α ∈S is majorized(in the sense of (1.1)) by a unique maximal element of S (see [11], Definition VII.5.13).This is essentially the property we are interested in, modulo the following flaw: an F-inverse semigroup with a zero element θ must be a semilattice (Example 1.4.1o below); thisis because the maximal element majorizing θ cannot otherwise be unique.

In view of theexamples we want to discuss, we slightly weaken the above requirement, as follows:3

1.1 Definition A unital inverse semigroup will be called an eF-inverse semigroup ifevery α ∈S which is not zero element for S is majorized by a unique maximal element.The set of maximal elements of an eF-inverse semigroup will be usually denoted by M.1.2 Example (Toeplitz inverse semigroup) The example which originally motivatedthese considerations is the following: let G be a group, and let P be a unital subsemigroupof G. For every x ∈G put:βx : {t ∈P | xt ∈P} ∋t −→xt ∈{s ∈P | x−1s ∈P}. (1.2)Clearly, βx belongs to the symmetric inverse semigroup IP.

Note that βx can be the voidmap θ on P; this happens if and only if x ∈G \ PP −1. Let SG,P be the subsemigroup ofIP generated by {βx | x ∈G}; this is a unital ⋆-subsemigroup, since β∗x = βx−1, x ∈G,and since βe (e = unit of G) is the identity map on P. Due to its relation to the Toeplitzalgebra associated to G and P, we shall call SG,P the Toeplitz inverse semigroup of (G, P).Now, if PP −1 = G, it turns out that SG,P has no zero element, and that it is an F-inverse semigroup, with M = {βx | x ∈PP −1}.

On the other hand, if PP −1 ̸= G, thenSG,P certainly has a zero element (the void map θ, which is βx for every x ∈G \ PP −1);in this case, SG,P still is an eF-inverse semigroup, where again M = {βx | x ∈PP −1} (seeSection 3 below).1.3 Example (singly generated inverse semigroup) Let S be a unital inverse semigroupwhich is generated (as unital ⋆-semigroup) by an element β ∈S. One checks immediately,by embedding S into a symmetric inverse semigroup, that we have βmβn ≤βm+n for everym, n ∈Z (where we make the convention that βn means β⋆|n| for n < 0).

Since any elementof S is a product of β’s and β∗’s, it immediately follows that for every α ∈S there exists(at least one) n ∈Z such that α ≤βn.In many examples (of S generated by β, as above), the n ∈Z such that α ≤βn isuniquely determined, for every α ∈S which is not zero element. This implies that S is aneF-inverse semigroup, with M = {n ∈Z | βn is not zero element for S}.More precisely, let us assume (without loss of generality) that S ⊆IT, the symmetricinverse semigroup on a non-void set T. Denote the subset ∩n∈ZDom(βn) of T by T∞, andput Tf = T \ T∞.

Then T∞and Tf are invariant for β, and β|T∞is a bijection. It iseasily seen that the only situation when S can fail to be an eF-inverse semigroup is when4

Tf, T∞̸= ∅, β|T∞is periodic and nonconstant, and β|Tf is nilpotent, i.e. there is a powerof it which is the void map.

This does not happen, for instance, in any of the examplesdiscussed in [4, 5].1.4 Examples Various examples of F-inverse semigroups are studied in the algebraictheory of inverse semigroups. For instance:1o Let E be a unital semilattice, i.e.

a unital inverse semigroup all the elements of whichare idempotents (see [11], I.3.9). Every α ∈E is selfadjoint (because α3 = α, hence theunique α∗satisfying αα∗α = α, α∗αα∗= α∗is α∗= α).

Hence α ≤β ⇔αβ = α, α, β ∈E,which implies that E is an eF-inverse semigroup with M = {ǫ}.Let us recall here the basic fact that any two idempotents of an arbitrary inverse semi-group S are commuting (see [11], II.1.2). This implies that, for every inverse semigroup S,the subset of idempotents S(o) = {α ∈S | α2 = α} is a subsemigroup ( a semilattice).2o Let G be a group, and let (Gn)n∈N be a sequence of subgroups of G, such that G0 = Gand Gn+1 ⊆Gn, n ≥0.

Consider also G∞= ∩n∈NGn. Define S = ∪0≤n≤∞Gn × {n}, withmultiplication(x, m)(y, n) = (xy, min(m, n)),0 ≤m, n ≤∞, x ∈Gm, y ∈Gn.This is an example of a Clifford E-unitary semigroup (see [11], II.2 and III.7.1).

It is anF-inverse semigroup, with M = (∪n∈N(Gn \ Gn+1) × {n}) ∪(G∞× {∞}).3o Let G be a group and let σ : G →G be an automorphism. Define S = N × G × N,with multiplication(m, x, n)(p, y, q) =m −n + max(n, p), σmax(n,p)−n(x)σmax(n,p)−p(y), q −p + max(n, p),for m, n, p, q ∈N and x, y ∈G.

This is an example of a Reilly semigroup (see [11], II.6); itis an F-inverse semigroup, with M = {(m, x, n) | m, n ∈N, x ∈G, min(m, n) = 0}. In thecase when G has only one element, S(≃N × N) is called the bicyclic semigroup; this wasalso appearing in the context of Example 1.3 (for (G, P) = (Z, N), in the notations usedthere).4o It should also be kept in mind that, obviously, every group is an F-inverse semigroup(for an F-inverse semigroup S we have “S group ⇔M = S”).1.5 The multiplicative structure on M Let S be an eF-inverse semigroup, and letM be the set of maximal elements of S. It is clear that M contains the unit ǫ of S, andthat it is closed under ⋆-operation.5

We shall work with a multiplicative structure on M, which is related to the multiplica-tion of S, but can’t of course coincide with it. In order to avoid any confusion, we shall usethe notationM = {βx | x ∈M},(1.3)where M is some fixed set of the same cardinality with M (and M ∋x →βx ∈M is abijection), and we shall define the multiplicative structure we need as an operation on M.We denote by e the unique element of M such that βe = ǫ (=unit of S).The case when S has not a zero element is quite clear, and well-known (see [11], III.5,VII.6).

We define the multiplication on M byx · y = the unique z ∈M such that βxβy ≤βz,x, y ∈M. (1.4)Then M becomes a group (e is the unit, and the inverse of x ∈M is the unique u ∈Msuch that βu = β∗x).

The map S →M which sends α ∈S into the unique x ∈M withβx ≥α is, clearly, a surjective semigroup homomorphism; moreover, it is easily seen thatevery homomorphism of S onto a group can be factored by it. Hence M is the maximalgroup homomorphic image of S.If S has a zero element θ, then we have on M only a partially defined multiplication,M(2) = {(x, y) ∈M × M | βxβy ̸= θ} −→M,(1.5)described by the same rule as in (1.4).

Still, for all our purposes this partial multiplicationwill be as reliable as a group structure (actually, it is not clear whether it wouldn’t bealways possible to embed it into a group). Remark that:- We still have a partial associativity property; i.e., if we putM(3) = {(x, y, z) ∈M × M × M | βxβyβz ̸= θ},(1.6)then (x · y) · z and x · (y · z) make sense and are equal for every (x, y, z) ∈M(3).- e still is a unit for M, i.e.

(x, e), (e, x) ∈M(2) and e · x = x · e = x for all x ∈M.- For every x ∈M, the unique u ∈M such that βu = β∗x, which will be denoted byx−1, has (x−1, x), (x, x−1) ∈M(2) and x−1 · x = x · x−1 = e. Moreover, it is immediatethat (x−1)−1 = x, for every x ∈M, and that (x, y) ∈M(2) ⇒(y−1, x−1) ∈M, y−1 · x−1 =(x · y)−1.- If (x, y) ∈M(2) and x · y = z, then automatically (x−1, z), (z, y−1) ∈M(2), andx−1 · z = y, z · y−1 = x. Indeed, we have βxβy ̸= θ ⇒βxβyβ∗y ̸= θ (since (βxβyβ∗y)βy =βxβy), hence (x·y)·y−1 and x·(y·y−1) make sense and are equal, by the partial associativity;6

but (x · y) · y−1 = z · y−1, while x · (y · y−1) = x. The equality x−1 · z = y is proved similarly.Note that, as a consequence, the partial multiplication on M has the cancellation property(x · y = x · z or y · x = z · x ⇒y = z).2The groupoid construction2.1 Definition Let S be a unital inverse semigroup, and let Ωbe a locally compactHausdorffspace.

By an action of S on Ωwe shall understand a ⋆-homomorphism Φ of Sinto the symmetric inverse semigroup of Ω, such that:(i) for every α ∈S, the domain Dom(Φ(α)) and the range Ran(Φ(α)) of Φ(α) are openin Ω, and Φ(α) is a homeomorphism between them;(ii) Φ(ǫ)) is the identity map on Ω(where ǫ denotes, as usual, the unit of S);(iii) if S has a zero element θ, then Φ(θ)) is the void map on Ω.For the terminology and basic facts about locally compact groupoids used in this note,we refer the reader to the monograph [12]. The following groupoid construction is a naturalgeneralization to eF-inverse semigroups of the very basic example of groupoid associated toa group action ([12], Example I.1.2a).2.2 Definition Let S be a unital eF-inverse semigroup, and let Φ be an action of Son the locally compact Hausdorffspace Ω.

We use the notations related to S which wereintroduced in Section 1.5 above (M such that M = {βx | x ∈M}, and the multiplicativestructure on M). We then define a groupoid G, as follows:G = {(x, ω) | x ∈M, ω ∈Dom(Φ(βx)) ⊆Ω}.

(2.1)The space of units of G is Ω, and the domain and range of (x, ω) ∈G are:d(x, ω) = ω, r(x, ω) = (Φ(βx))(ω). (2.2)The multiplication on G is defined by the rule:(x, ω)(x′, ω′) = (x · x′, ω′),(2.3)7

for (x, ω), (x′, ω′) ∈G such that (Φ(βx′))(ω′) = ω. (Note: from the latter equality andω ∈Dom(Φ(βx)) it follows that ω′ is in the domain of Φ(βx)Φ(βx′) = Φ(βxβx′); thenβxβx′ can’t be zero element for S, and using βxβx′ ≤βx·x′ we get ω′ ∈Dom(Φ(βx·x′)),(Φ(βx·x′))(ω′) = (Φ(βx))(ω).Thus the right-hand side of (2.3) is indeed in G, and hasd(x · x′, ω′) = d(x′, ω′), r(x · x′, ω′) = r(x, ω).

)The identity at the unit ω ∈Ωis (e, ω), with e the unit of M, and the inverse of(x, ω) ∈G is (x−1, (Φ(βx))(ω)).The topology on G (⊆M × Ω) is the one obtained by restricting to G the product ofthe discrete topology on M and the given topology on Ω. This is locally compact (andHausdorff), since G is obviously open in M × Ω.

It is immediately seen that multiplicationand taking the inverse on G are continuous, i.e.G is a locally compact groupoid. Thetopology induced from G to the space of identities G(o) = {(e, ω) | ω ∈Ω} ≃Ωis the one ofΩ.

Moreover, G(o) is open in G, i.e. G is an r-discrete groupoid ([12], Definition I.2.6).2.3 Remark Note that G in 2.2 is the disjoint union of the sets {x}×Dom(Φ(βx)), x ∈M, each of them open and closed in G. For every f ∈Cc(G), supp f is contained in a finiteunion ∪ni=1{xi} × Dom(Φ(βxi)), and we can write f = Pni=1 fhi with hi the characteristicfunction of {xi} × Dom(Φ(βxi)), 1 ≤i ≤n (fhi ∈Cc(G), since hi is continuous).

Thisshows that we have the direct sum decompositionCc(G) = ⊕x∈M{f ∈Cc(G) | supp f ⊆{x} × Dom(Φ(βx)) }. (2.4)Note also that from Proposition I.2.8 of [12] it follows that the counting measures forma Haar system on G (in the sense of [12], Definition I.2.2).2.4 Remark Let S be a unital inverse semigroup generated by an element β ∈S, andsuch that (as in Example 1.3 above), every α ∈S which is not zero element is majorizedby a unique βn, n ∈Z.

We have an obvious choice of notation for the set M appearingin equation (1.3) of 1.5. If βn is not zero element for S, for every n ∈Z, then S has nozero element, M = {βn | n ∈Z}, and we take M = Z (the map M →M being n →βn).Otherwise, S must have a zero element θ, and there exists m ≥1 such that βm ̸= θ = βm+1,β⋆m ̸= θ = β⋆(m+1); we then have M = {βn | |n| ≤m}, and we take M = Z ∩[−m, m].In any case, the (possibly partially defined) multiplication on M is the usual addition ofintegers.

For Φ an action of S on a space Ω, as in 2.1, note that the groupoid G associated8

to Φ isG = {(m, ω) | m ∈Z, ω ∈Dom(Φ(βn))},(2.5)even in the case when M is of the form Z ∩[−m, m] (in this case we have Dom(Φ(βn)) = ∅for |n| > m).Now, remark that G of (2.5) is an r-discrete locally compact groupoid (with the struc-ture defined in 2.2), even if there is no assumption on β to ensure that S is aneF-inverse semigroup. Indeed, it is seen immediately that the only things required for havinga valid groupoid structure on G of (2.5) are the inequality βnβm ≤βn+m, n, m ∈Z, andthe group properties of the addition of the integers, which hold unconditionally.Moreover, the isomorphism between the C*-algebra of G in (2.5), on one hand, and thecrossed-product C*-algebra (in the sense of [4]) by the partial automorphism given by β, onthe other hand, will also turn out to hold with no condition on β (see Theorem 4.1 below).The above considerations suggest an other simple method of constructing groupoidsassociated to actions of inverse semigroups.

Let G be a group, let S be a unital inversesemigroup, and let G ∋x →βx ∈S be a map such that: βe = ǫ, where e is the unit of Gand ǫ the one of S; βx−1 = β∗x for every x ∈G; βxβy ≤βxy for every x, y ∈G; and for everyα ∈S there exists (at least one) x ∈G such that α ≤βx. (In other words, instead of givingconditions which ensure a multiplicative structure on a remarkable family of elements of S,we impose this from outside.) Then, to an action Φ of S on a locally compact space Ω, onecan associate the locally compact groupoidG = {(x, ω) | x ∈G, ω ∈Dom(Φ(βx))},(2.6)with groupoid structure defined as in 2.2.

Note that the actions of Toeplitz inverse semi-groups can also be considered in this way (in the notations of Example 1.2, we takeG ∋x →βx ∈SG,P to be exactly the one given by equation (1.2)); however, the propertiesshown in Section 6 below don’t hold in general for groupoids of the type (2.6).3Example: the Toeplitz inverse semigroupWe now return to the framework of Example 1.3. Consider G, P, (βx)x∈G and SG,P ⊆IPas in 1.3.

We begin by proving the assertions which were made there about SG,P .3.1 Lemma If PP −1 = G, then SG,P has no zero element.9

Proof If SG,P has a zero element θ, then this must be the void map on P (we leavehere apart the case when G = P = {e}).Indeed, θ would otherwise be the identicaltransformation of a subset of P which is fixed by every βx, x ∈PP −1; but for x ̸= e, βxhas no fixed points.Hence, since an arbitrary element of SG,P is a product of βx’s, what we need to show isthat for every n ≥1 and x1, . .

. , xn ∈G, the partially defined transformation βx1 · · · βxn onP is non-void.Let x1, .

. .

, xn ∈G be arbitrary. Since PP −1 = G, we can find s1, .

. .

, sn, t1, . .

. , tn ∈Psuch that: xn = snt−1n , xn−1sn = sn−1t−1n−1, xn−2sn−1 = sn−2t−1n−2, .

. ., x1s2 = s1t−11 .Then t = tn .

. .

t1 ∈P is in the domain of βx1 · · · βxn, because, as it is easily checked,xj . .

. xnt = sjtj−1 .

. .

t1 ∈P, 1 ≤j ≤n. Thus βx1 · · · βxn is indeed non-void.

QED3.2 Lemma For every non-void α ∈SG,P there exists a unique x ∈PP −1 such thatα ≤βx; this x can be expressed as α(t)t−1, with t arbitrary in the domain of α.Proof Write α = βx1 · · · βxn, with x1 · · · xn ∈PP −1; then Dom(α) = {t ∈P |xnt, xn−1xnt, . .

. , x1 · · · xnt ∈P}, and α(t) = x1 · · · xnt for t ∈Dom(α).Putting x =x1 · · · xn, we get α(t)t−1 = x for every t ∈Dom(α) (in particular x is in PP −1).

It is clearthat α ≤βx, and that x is the unique element of PP −1 having this property. QEDFrom the above two lemmas it is clear that no matter if PP −1 = G or not, SG,P isan eF-inverse semigroup, with M = {βx | x ∈PP −1}.

It fits very well the notations ofSection 1.5 to take M = PP −1. Note also that the multiplication defined on M = PP −1 byequation (1.4) of 1.5 coincides with the one coming from the group G; this happens becausefor any x, y ∈PP −1 such that βxβy ̸= θ, βxβy acts on its domain by left translation withxy.We now pass to describe a remarkable action of SG,P, which gives via the constructionof 2.2 the Wiener-Hopf groupoid G, introduced by P.Muhly and J.Renault in [7].

The spaceof the action of SG,P will beΩ=clos{tP −1 | t ∈P} ⊆{0, 1}G,(3.1)where {0, 1}G is identified to the space of all subsets of G.The topology on Ωis the10

restriction of the product topology on {0, 1}G, and is compact and Hausdorff. Note thatP −1 ⊆A ⊆PP −1 for every A ∈Ω.3.3 Proposition For every α ∈SG,P we define:(Φ(α) : clos{tP −1 | t ∈Dom(α)} →clos{sP −1 | s ∈Ran(α)},(Φ(α))(A) = xA,(3.2)where x in (3.2) is the unique element of PP −1 such that α ≤βx.

(If α = θ, the void mapon P, we take by convention Φ(α) to be the void map on Ω.) Then Φ(α) makes sense forevery α ∈SG,P, and Φ is an action of SG,P on Ω, in the sense of Definition 2.1.Proof Let α ∈SG,P be non-void, and let x be the unique element of PP −1 such thatα ≤βx.

The map A →xA (={xa | x ∈A}) is continuous on {0, 1}G, hence {A ⊆G | xA ∈clos{sP −1 | s ∈Ran(α)} } is closed. This set contains tP −1 for every t ∈Dom(α) (becauset ∈Dom(α) ⇒xt = α(t) ∈Ran(α), and on the other hand x(tP −1) = (xt)P −1); so Φ(α)defined in (3.2) makes sense.

It is also clear that Φ(α∗) is an inverse for Φ(α), which is thusa bijection.It is useful to note that if α ∈SG,P is written as a product βx1 · · · βxn (for some n ≥1and x1, . .

. , xn ∈PP −1), then we have the equivalent characterizationDom(Φ(α)) = {A ∈Ω| A−1 ∋xn, xn−1xn, .

. .

, x1 · · · xn}(3.3)(note:this is valid even if α = θ).Indeed, the right-hand side of (3.3) is immedi-ately seen to be equal to clos{tP −1 | (tP −1)−1 ∋xn, xn−1xn, . .

. , x1 · · · xn} = clos{tP −1 |t, xnt, xn−1xnt, .

. .

, x1 · · · xnt ∈P}, which is exactly Dom(Φ(α)).As a consequence of (3.3), it is clear that Dom(Φ(α)) (and Ran(Φ(α)) = Dom(Φ(α∗)),too) is open in Ωfor every α ∈SG,P .We are left to show that Φ(α)Φ(α′) = Φ(αα′) for every α, α′ ∈SG,P. If one of α, α′ is θ,then both Φ(α)Φ(α′) and Φ(αα′) are the void map on Ω, so we shall assume α ̸= θ ̸= α′ (butwe don’t assume αα′ ̸= θ).

We take x1, . .

. , xm, y1, .

. .

, yn ∈PP −1 such that α = βx1 · · · βxm,α′ = βy1 · · · βyn; note that the unique x, y ∈PP −1 such that α ≤βx, α′ ≤βy must then bex = x1 · · · xm and y = y1 · · · yn. Using (3.3) we see thatDom(Φ(α)Φ(α′)) =A ∈Ω| A−1 ∋yn, yn−1yn, .

. .

, y1 · · · yn,(yA)−1 ∋xm, xm−1xm, . .

. , x1 · · · xm11

=A ∈Ω| A−1 ∋yn, yn−1yn, . .

. , y1 · · · yn,xmy1 · · · yn, xm−1xmy1 · · · yn, .

. .

, x1 · · · xmy1 · · · yn= Dom(Φ(αα′))(where the latter equality holds also by (3.3), since αα′ = βx1 · · · βxmβy1 · · · βyn). If Dom(Φ(α)Φ(α′)) = Dom(Φ(αα′)) is non-void, it is clear that both transformations act on it by lefttranslation with xy, hence they are equal.

QEDNote that if P ∩P −1 = {e}, then Ωof (3.1) is a compactification of P (by identifyingt ∈P with tP −1); in some sense, Ωis obtained from P by adding one point for each “typeof convergence to ∞in P” (compare to the comments in Section 2.3.1 of [8]).Takinginto account that a non-void α in SG,P is actually the left translation with x on Dom(α),with x ∈PP −1 such that α ≤βx, we can interprete the map Φ(α) of (3.2) as a sort of“compactification of α”. 1Finally (without having to assume P ∩P −1 = {e}), we have, in view of the characteri-zation (3.3):Dom(Φ(βx)) = {A ∈Ω| A ∋x−1},x ∈PP −1.

(3.4)Hence the groupoid associated as in 2.2 to the above action of SG,P on Ωis:G = {(x, A) | A ∈Ω, x ∈A−1}. (3.5)This is exactly the form given to the Wiener-Hopf groupoid in [8], Proposition 2.3.4.4Example: singly generated inverse semigroupsLet S be a unital inverse semigroup, generated (as unital ⋆-semigroup) by an elementβ ∈S, and let Φ be an action of S on the locally compact space Ω(as in 2.1).

We denoteΦ(β) by eβ. Let G = {(n, ω) | n ∈Z, ω ∈Dom( eβn)} be the groupoid associated to thisaction, as in equality (2.5) of Remark 2.4.1 One may ask what happens if we don’t do any compactification, and just take the obvious action ofSG,P on P. It is immediate that the groupoid associated to this would be the total equivalence relation onP, having thus C*-algebra isomorphic to the compact operators on l2(P).12

On the other hand, if we consider the ideals I = {f ∈Co(Ω) | f ≡0 on Ω\ Ran( eβ)},J = {f ∈Co(Ω) | f ≡0 on Ω\ Dom( eβ)} of Co(Ω), then eβ determines an isomorphismθ : I →J,(θ(f))(ω) =(f( eβ(ω)),if ω ∈Dom( eβ)0,otherwise, for f ∈I.Thus Θ = (θ, I, J) is a partial automorphism of Co(Ω), in the sense of R. Exel [4], Definition3.1, and has attached to it a covariance C*-algebra C*(Co(Ω), Θ) (see [4], Definition 3.7).4.1 Theorem Assuming Ωsecond countable, we have that C*(Co, Θ) and C*(G) arenaturally isomorphic.Proof Following the notations of [4], Section 3, let us putDn = {f ∈Co(Ω) | f ≡0 on Ω\ Dom( eβn)},n ∈Z;in other words, Dn is the domain of θ−n. We have in particular Do = Co(Ω), D1 = J,D−1 = I.

Let L be the vector space of sequences (fn)n∈Z, with fn ∈Dn for every n, andsuch that Pn∈Z ||fn|| < ∞; for m ∈Z and f ∈Dm denote by fδm the sequence (fn)n∈Zin L which has fm = f and fn = 0 for n ̸= m. Then L is given (in [4], p.7) a ⋆-algebrastructure which in particular has for n, m ∈Z, fn ∈Dn, fm ∈Dm: (fnδn) ⋆(fmδm) =gδn+m, (fnδn)∗= hδ−n, with:g(ω) =(fn(ω)fm( eβn(ω)),if ω ∈Dom( eβn) ∩Dom( eβn+m)0,otherwise,(4.1)h(ω) =(fn( eβ−n(ω)),if ω ∈Dom( eβ−n)0,otherwise. (4.2)C*(Co(Ω), Θ) is defined as the enveloping C*-algebra of L, with respect to the norm||(fn)n∈Z||1 = Pn∈Z ||fn||.For every n ∈Z let us definedDn = {f ∈Cc(Ω) | supp f ⊆Dom( eβn)} ⊆Dn,and let bL be the space of finitely supported sequences (fn)n∈Z with the property thatfn ∈dDn for every n. Then clearly bL is a ⋆-subalgebra of L, dense in ||·||1, and C*(Co(Ω), Θ)can also be defined as the enveloping C*-algebra of (bL, || · ||1).

Since Ωis assumed to besecond countable, the latter normed ⋆-algebra is separable, and we may consider only itsrepresentations on separable Hilbert spaces.13

Now, every ⋆-representation π : bL →L(H) is automatically contractive with respectto || · ||1. In order to verify this, it clearly suffices to check that ||π(fnδn)|| ≤||fn|| forevery n ∈Z and fn ∈dDn.And indeed, one sees immediately (from (4.1),(4.2)) that(fnδn) ⋆(fnδn)∗= |fn|2δo, hence ||π(fnδn)||2 = ||π(|fn|2δo)||; the latter quantity does notexceed || |fn|2 || = ||fn||2, because π restricted to {fδo | f ∈Cc(Ω)} ⊆bL gives a ⋆-representation of Cc(Ω), which is automatically contractive.

(The latter assertion holdseven if Ωis non-compact, due to the fact that, for every f ∈Cc(Ω), the spectral radius off in the unitization of Cc(Ω) equals ||f||. )It results that C*(Co(Ω), Θ) is the enveloping C*-algebra of bL, considered with respectto all the algebraic ⋆-representations of bL on separable Hilbert spaces.On the other hand, due to the separability condition imposed on Ω, it is clear that G issecond countable; hence, by Corollary II.1.22 of [12] (see also Corollaire 4.8 of [13]), C*(G)is the enveloping C*-algebra of Cc(G) with respect to all its (algebraic) ⋆-representationson separable Hilbert spaces.

This makes clear that the Theorem will follow as soon as wecan prove that the ⋆-algebras bL and Cc(G) are isomorphic.For every n ∈Z and fn ∈dDn let us denote by χn ⊗fn the restriction to G ⊆Z × Ωofthe direct product between χn = (characteristic function of {n}) and fn. We denote, forevery n ∈Z, Dn = {χn ⊗fn | fn ∈dDn} ⊆Cc(G).

From the direct sum decomposition ofequation (2.4) it is immediate that Cc(G) ≃⊕n∈ZDn. (Note that some of the spaces Dnmay be reduced to {0}, if eβ is nilpotent; in this case, the corresponding Dn’s are also {0}.

)In view of the obvious decomposition bL = ⊕n∈Z{fnδn | fn ∈dDn}, it becomes clear that bLand Cc(G) are naturally isomorphic as vector spaces.Recalling the definition of the multiplication and ⋆-operation on Cc(G) (from [12], Propo-sition II.1.1) one gets, for m, n ∈Z, fn ∈dDn, fm ∈dDm, the formulae: (χn ⊗fn)⋆(χm ⊗fm)= χn+m ⊗g, (χn ⊗fn)∗= χ−n ⊗h, where:g(ω) =(fn( eβm(ω))fm(ω),if ω ∈Dom( eβm) ∩Dom( eβn+m)0,otherwise,(4.3)and h is the same as in (4.2).Taking into account the difference of choice which appears in the definition of the mul-tiplication in the two approaches (i.e. in (4.1) vs (4.3)), one has thus to proceed as follows:for every n ∈Z and fn ∈dDn denote by Γn(fn) the complex conjugate of the function in(4.2).

Then the linear isomorphism bL →Cc(G) determined by fnδn →χ−n ⊗Γn(fn), n ∈Z,fn ∈dDn, is also an isomorphism of ⋆-algebras (the easy verification of this is left to thereader). QED14

4.2 Remark Crossed-products by partial isomorphisms were used in [5] to approachAF-algebras, and in particular to approach in a very explicit way UHF-algebras. This leadsto an interesting point of view on the Glimm groupoid (and more generally on AF-groupoids,defined in [12], Section III.1), as being close, in some sense, to transformation groups.More precisely, let (ni)∞i=0 be a sequence of positive integers, and let X = Q∞i=0{0, 1, .

. .,ni −1} have the product topology.

Consider, as in [5], the “restricted odometer map” β∗: X \ {(n1 −1, n2 −1, . .

.)} →X \ {(0, 0, .

. .)} which is given by the formal addition with(1, 0, 0, .

. .

), with carry over to the right. Let β be the inverse of β∗, let S be the unital⋆-semigroup generated by β in IX, and let G be the groupoid associated to the actionΦ(α) ≡α of S on X.

Since β is very close to be a homeomorphism of X, G is in some senseclose to be a transformation group. On the other hand, G is easily seen to be isomorphic tothe Glimm groupoid on (ni)∞i=0 (defined in [12], p.128); the fact that C*(G) ≃UHF((ni)∞i=0)can be verified either this way, or by combining the above Theorem 4.1 with Theorem 3.2of [5].5The relation with localizations (in the sense of Kumjian)A. Kumjian ([6]) has developed a method which associates a C*-algebra to an inversesemigroup S of homeomorphisms between open subsets of a locally compact space Ω, suchthat:(Dom(α))α∈S is a basis for the topology of Ω.

(5.1)Following [6] Definition 2.3, such an inverse semigroup will be called a localization.The point of view of this note is slightly different from the one of [6]; for instance, theapproach to the Glimm groupoid mentioned in Remark 4.2 above is different from the onetaken in [6] (see 5.2 below); actually, the inverse semigroup generated by the restrictedodometer map is not a localization, and so is also the case for the example described inSection 3 (even in the classical situation when (G, P) = (Z, N) ).Still, the C*-algebra construction of [6] coincides with the C*-algebra of the groupoiddefined in 2.2, on the common territory of the two approaches:5.1 Theorem Let S be a countable eF-inverse semigroup of homeomorphisms between15

open subsets of the locally compact space Ω, and assume that S is a localization. Let Gbe the groupoid associated (as in 2.2) to the action Φ(α) ≡α of S on Ω.

Then C*(G) isisomorphic to the C*-algebra associated to S in [6].Proof The C*-algebra associated to S in 1 [6] is the envelopation of a ⋆-algebra whichwe will denote (following [6], Section 5) by Cc(S). An argument similar to the one usedin the proof of Theorem 4.1 shows that it suffices to prove the isomorphism of ⋆-algebrasCc(S) ≃Cc(G).We now have to go into the details of the definition of Cc(S).

Following [6], Notation5.2, let us putD(S) = ⊕α∈S{(α, f) | f ∈Cc(Dom(α))},(5.2)where Cc(Dom(α)) stands for {f ∈Cc(Ω) | supp f ⊆Dom(α)}, and where the summand in(5.2) corresponding to α is given the linear structure coming from Cc(Dom(α)). For α ∈Sand f ∈Cc(Dom(α)) we shall view (α, f) as an element of D(S), in the obvious manner.D(S) is given (in [6], 5.2) a ⋆-algebra structure, such that for α1, α2 ∈S, f1 ∈Cc(Dom(α1)),f2 ∈Cc(Dom(α2)) we have (α1, f1)(α2, f2) = (α1α2, g), (α1, f1)∗= (α∗1, h), with:g(ω) =(f1(α2(ω))f2(ω),if ω ∈Dom(α2)0,otherwise ,(5.3)h(ω) =(f1(α∗1(ω)),if ω ∈Dom(α∗1)0,otherwise .

(5.4)Then Cc(S) is D(S)/Io(S) ([6], p.160), where the definition of the ideal Io(S) ⊆D(S)remains to be recalled.On the other hand, let us also consider the groupoid G = {(x, ω) | x ∈M, ω ∈Dom(βx)}defined in 2.2, where {βx | x ∈M} is the set of maximal elements of S, as in 1.5. For everyx ∈M and f ∈Cc(Dom(βx)) let us denote by χx⊗f ∈Cc(G) the restriction to G(⊆M ×Ω)of the direct product between χx = (characteristic function of x) and f.For α ∈S which is not zero element, consider the unique x ∈M such that α ≤βx;then Dom(α) ⊆Dom(βx), hence we have a linear map (α, f) →χx ⊗f from {(α, f) | f ∈Cc(Dom(α))} into Cc(G).

The direct sum (after α ∈S) of all these linear maps is a linearmap D(S) →Cc(G), which will be denoted by ρ. Comparing (5.3),(5.4) with the definitionof the operations on Cc(G) (given by formulae similar to (4.3),(4.2) of Section 4), one checks1 We warn the reader about the unfortunate coincidence that Ωis used in the present paper to denote aspace, and in [6] to denote an inverse semigroup.16

immediately that ρ is actually a ⋆-algebra homomorphism. ρ is clearly onto, and we are leftto show that Ker ρ equals Io(S), the ideal of D(S) mentioned following to (5.4).The definition of Io(S) depends on the following notion: a finite family (αa, fa)a∈A (withαa ∈S, fa ∈Cc(Dom(αa)) for a ∈A) is called coherent if there are open subsets (Ua)a∈Aof Ωsuch that:(supp fa ⊆Ua ⊆Dom(αa),a ∈A,αa|Ua ∩Ua′ = αa′|Ua ∩Ua′ for a, a′ ∈A having Ua ∩Ua′ ̸= ∅.

(5.5)Using this notion, one arrives to Io(S) in several steps:- Define I(S) to be the linear span of {Pa∈A(αa, fa) | (αa, fa) coherent, Pa∈A fa = 0};I(S) is shown to be a two-sided, selfadjoint ideal of D(S) ([6], p. 158).- For every ξ ∈D(S) consider the number|ξ|′o = infXb∈B||Xa∈Abfa,b||∞,where the infimum is taken after all the decompositions ξ = Pb∈BPa∈Ab(αa,b, fa,b) (withB and (Ab)b∈B finite sets), such that each of the families (αa,b, fa,b)a∈Ab (b ∈B) is coherent.- For every ξ ∈D(S) consider the number|ξ|o = inf η∈I(S)|ξ −η|′o;then | · | is a ⋆-algebra seminorm on D(S) ([6], p.159).- Define Io(S) = {ξ ∈D(S) ; |ξ|o = 0}.An obvious situation of coherent family (αa, fa)a∈A can be obtained by taking all theαa’s (a ∈A) to be majorized by the same βx, x ∈M; we shall call such a family majorized-coherent. It is immediate that an element ξ ∈D(S) belongs to Ker ρ if and only if itis of the form Pni=1Pa∈Ai(αa, fa), where each of the families (αa, fa)a∈Ai (1 ≤i ≤n) ismajorized-coherent.

This makes clear that Ker ρ ⊆I(S); the opposite inclusion is alsotrue, as implied by the following5.1.1 Lemma If (αa, fa)a∈A is a coherent family, with Pa∈A fa = 0, then there existsa partition A = A1 ∪. .

. ∪An of A such that each of the families (αa, fa)a∈Aj (1 ≤j ≤n)is majorized-coherent with Pa∈Aj fa = 0.Proof of Lemma 5.1.1 Without loss of generality, we may assume that αa is non-zero for every a ∈A; then, for every a ∈A, we can consider the unique x(a) ∈M such17

that αa ≤βx(a), and we can partition A such that a, a′ are in the same component of thepartition if and only if x(a) = x(a′).Pick a family (Ua)a∈A of open subsets of Ωsuch that (5.5) holds. We will show thatx(a) ̸= x(a′) ⇒Ua ∩Ua′ = ∅; this immediately implies that the partition considered in thepreceding paragraph satisfies the required conditions.So, let a, a′ be in A such that Ua ∩Ua′ ̸= ∅.Using (5.1), we can find a non-zeroidempotent γ ∈S such that Dom(γ) ⊆Ua ∩Ua′.

Then using (5.5) we get αaγ = αa′γ,non-zero. We have αaγ ≤βx(a), αa′γ ≤βx(a′); but the maximal element of S majorizingαaγ = αa′γ is unique, hence x(a) = x(a′).We have thus obtained I(S) = Ker ρ.

It is obvious that ξ ∈I(S) ⇒|ξ|′o = 0 ⇒|ξ|o = 0,hence Io(S) ⊇I(S). On the other hand, the inclusion Io(S) ⊆Ker ρ will come out fromthe following5.1.2 Lemma: If ξ = Pb∈BPa∈Ab(αa,b, fa,b) ∈D(S) with (αa,b, fa,b)a∈Ab coherent forevery b ∈B, and if x ∈M, ω ∈Dom(βx), then|(ρ(ξ))(x, ω)| ≤Xb∈B|Xa∈Abfa,b(ω)|.

(5.6)Indeed, let us assume the Lemma 5.1.2 true. Fixing ξ and (αa,b, fa,b)a,b in (5.6), andletting (x, ω) run in G, gives:||ρ(ξ)||∞≤Xb∈B||Xa∈Abfa,b||∞.

(5.7)Then keeping ξ fixed, but taking in (5.7) the infimum after all the possible writings ξ =Pb∈BPa∈Ab(αa,b, fa,b), we obtain||ρ(ξ)||∞≤|ξ|′o. (5.8)Finally, replacing in (5.8) ξ by ξ −η, with η ∈I(S), using that ρ(η) = 0, and taking anotherinfimum, leads us to the inequality||ρ(ξ)||∞≤|ξ|o;(5.9)this makes the inclusion Io(S) ⊆Ker ρ clear.So we are left to make the18

Proof of Lemma 5.1.2 For every b ∈B, denote {a ∈Ab | αa,b ≤βx} by A′b, andAb\A′b by A′′b . By the definition of ρ we have (ρ(ξ))(x, ω) = Pb∈BPa∈A′b fa,b(ω), which gives|(ρ(ξ))(x, ω)| ≤Pb∈B | Pa∈A′b fa,b(ω)|.

It remains to pick an element b of B and verify that| Pa∈A′b fa,b(ω)| ≤| Pa∈Ab fa,b(ω)|; this is clear when Pa∈A′b fa,b = 0, so we shall assumethat there exists at least one a ∈A′b such that fa,b(ω) ̸= 0. But, by an argument similarto the one proving Lemma 5.1.1, it is seen that (∪a∈A′b supp fa,b) ∩(∪a∈A′′b supp fa,b) = ∅.Hence if there exists a ∈A′b such that fa,b(ω) ̸= 0, then we must have fa,b(ω) = 0 for alla ∈A′′b , so that | Pa∈A′b fa,b(ω)| = | Pa∈Ab fa,b(ω)|.

QED5.2 Remark In Section 3.6 of [6], a class of localizations with associated AF C*-algebrasis constructed. In particular, for a given sequence (ni)∞i=0 of positive integers, the followinglocalization L belonging to this class has UHF((ni)∞i=0) as associated C*-algebra:- the space on which L acts is X = Q∞i=0{0, 1, .

. .

, ni −1};- L itself is described as{γ(uo, . .

. , uk; vo, .

. .

, vk) | k ≥0, 0 ≤uj, vj ≤nj −1 for 0 ≤j ≤k},(5.10)where for uo, . .

. , uk, vo, .

. .

, vk as above the domain of γ(uo, . .

. , uk; vo, .

. .

, vk) is {(wo, w1,w2, . .

.) ∈X | wj = vj for j ≤k}, its range is {(wo, w1, w2, .

. .) ∈X | wj = uj for j ≤k}, andγ(uo, .

. .

, uk; vo, . .

. , vk) acts by replacing the first k + 1 components of the sequence.It is immediate that L of (5.10) is an eF-inverse semigroup (its maximal elements arethose in (5.10) having uk ̸= vk); hence Theorem 5.1 applies.

The groupoid G associated asin 2.2 to the action of L on X is easily seen to be (again) the Glimm groupoid. On theother hand, it is obvious that L is not singly generated, and that (although the acted spaceX is the same as in Remark 4.2) the intersection between L and the ⋆-semigroup generatedby the restricted odometer is reduced to {ǫ, θ}.Thus actions of two rather different inverse semigroups can give raise to the samegroupoid (and hence the same C*-algebra).

In the present case, the reason which makesthis happen is that both the inverse semigroup constructions, and the groupoid one, relyon the same method of finding m.a.s.a.’s in AF-algebras ([14], Section 1.1).5.3 Remark A localization S on the space Ωis called free ([6], Definition 7.1) if forevery α ∈S and ω ∈Dom(α) such that α(ω) = ω, there exists a neighborhood U of ω in Ωon which α acts as the identity. If in addition S is assumed to be an eF-inverse semigroup,19

it follows that every α ∈S which has a fixed point is an idempotent. Indeed, α, ω, U beingas above, we may assume (by using (5.1)) that the characteristic function χ of U belongsto S. From χ ≤α we infer that χ and α are majorized by the same maximal element of S,which can only be ǫ, the unit; but α ≤ǫ means that α is idempotent.As an immediate consequence, if S is a free localization on Ωand also an eF-inversesemigroup, then the groupoid G associated to S as in 2.2 is principal ([12], DefinitionI.1.1.1).

It is actually clear that G coincides with the equivalence relation “ω ∼ω′ def⇔thereis α ∈S such that α(ω) = ω′” on Ω; hence G is exactly as in Section 7.2 of [6].5.4 Example An example of localization which is an eF-inverse semigroup, but is notfree, is the one described in Section 10.1 of [6], which is related to the Cuntz-Kriegeralgebras.Recall from [2] that for an n × n matrix A with entries Ai,j ∈{0, 1}, 1 ≤i, j ≤n, whichis irreducible (in the sense that for every i, j there exists m ∈N such that (Am)i,j > 0), andis not a permutation matrix, there exists a unique C*-algebra OA generated by n non-zeropartial isometries S1, . .

. , Sn satisfying((SiS∗i )(SjS∗j ) = 0,for i ̸= jS∗i Si = Pnj=1 Ai,j(SjS∗j ),for 1 ≤i ≤n.

(5.11)For the matrix A as above, consider the compact space of A-admissible sequences,XA = {(jo, j1, j2, . .

.) | 1 ≤jk ≤n and Ajk,jk+1 = 1 for every k ≥0}(with topology obtained by restricting the product topology of {0, 1, .

. .

, n}N ). Then forevery 1 ≤m ≤n denote by βm the map:({(ik)k≥0 ∈XA | Am,io = 1} →{(jk)k≥0 ∈XA | jo = m}(io, i1, i2, .

. .) →(m, io, i1, i2, .

. .

);and let SA be the unital ⋆-semigroup of homeomorphisms between open compact subsets ofXA which is generated by β1, . .

. , βn.

SA is a localization; indeed, for every finite sequencejo, . .

. , jp such that Ajo,j1 = .

. .

= Ajp−1,jp = 1, the domain of β∗jp · · · β∗jo is the set Vjo,...,jpof sequences in XA beginning with jo, . .

. , jp, and the Vjo,...,jp’s are a basis of XA.

Notemoreover that the domains of β∗1, . .

. , β∗n form a partition of XA, on which β∗1, .

. .

, β∗n arethe restrictions of the one-sided shift on XA; hence SA is as in 10.1 of [6].On the other hand, it is not difficult to check that SA is an eF-inverse semigroup; weleave to the reader the verification of the following facts:20

5.4.1 Lemma Let Fn be the free group on generators g1, . .

. , gn, and let MA be the setof words x = gi1 · · · gipg−1jq · · · g−1j1 ∈Fn, with p, q ≥0 and 1 ≤i1, .

. .

, ip, j1, . .

. , jq ≤n, suchthat:(i) ip ̸= jq (i.e.

the word x is in reduced form);(ii) Ai1,i2, . .

. , Aip−1,ip, Aj1,j2, .

. .

, Ajq−1.jq are all 1;(iii) the set {1 ≤l ≤n | Aip,l = Ajq,l = 1} is not void.For every x = gi1 · · · gipg−1jq · · · g−1j1 ∈MA denote βi1 · · · βipβ∗jq · · · β∗j1 ∈SA by βx. Thenthe set of maximal elements of SA is {βx | x ∈MA}.

Moreover, the multiplicative structureon MA defined as in Section 1.5 above coincides with the one coming from Fn (i.e. forx, y ∈MA such that βxβy ̸= θ, the product in Fn of x and y is still in MA, and βxβy ≤βxy).From Theorem 5.1 above and from 10.1 of [6] it follows that the groupoid GA associated(as in 2.2) to SA has C*(GA) ≃OA.

Another proof of this isomorphism can be done by usingthe surjective homomorphism Ψ : C*(SA) →C*(GA) offered by Corollary 6.4 below: sinceβ1, . .

. , βn generate SA, the partial isometries Ψ(β1), .

. .

, Ψ(βn) generate C*(GA), and it iseasily checked that the latter ones satisfy (5.11) (thus C*(GA) ≃OA, due to the uniquenessof OA).It should be noted that if the matrix A has all the entries equal to 1, then the groupoidGA in the preceding paragraph coincides with the Cuntz groupoid of [12], Section III.2.1.6The homomorphism C*(S)→C*(G)Let S be an eF-inverse semigroup,and let Φ : S →IΩbe an action of S on a space Ω,as in 2.1, having the additional property that Dom(Φ(α)) is compact for every α ∈S. (This is true for the examples discussed in Sections 3, 4.2, 5.4.) Note that in particularΩ= Dom(Φ(ǫ)) must be compact.Consider the groupoid G associated to the action Φ (as in 2.2).The partition G =∪x∈M{x} × Dom(Φ(βx)) remarked in 2.2 consists now of open and compact subsets; eachof these subsets is a G-set ( which means that the restriction to it of the domain map andof the range map are one-to-one - see [12], Definition I.1.10).Recall now from [12], Definition I.2.10, that the open compact G-sets of G form (withthe pointwise multiplication) an inverse semigroup, called the ample semigroup of G. Thepartition mentioned in the preceding paragraph (which is indexed by the set M of maximal21

elements of S) “extends” to a homomorphism of inverse semigroups as in the followingLemma, the straightforward proof of which is left to the reader.6.1 Lemma For every α ∈S defineA = {x} × Dom(Φ(α)) ⊆G,(6.1)where x is the unique element of M such that α ≤βx. (If α has a zero element θ, we takeby convention A(θ) = ∅, the void set.) Then α →A(α) is a unital ⋆-homomorphism fromS into the ample semigroup of G.Now, it is a basic fact that if A and B are open compact G-sets of an r-discrete groupoidG, and if χA, χB, χAB are the characteristic functions of A, B, AB, respectively, then χABis the convolution of χA and χB in Cc(G), and moreover χ∗A = χA−1 in Cc(G).As aconsequence, Lemma 6.1 actually gives a unital ⋆-homomorphism S →Cc(G) ⊆C∗(G).

Thisextends by linearity to the “⋆-semigroup algebra” C[S]. (Recall that C[S] is the ⋆-algebrahaving S (or S \{θ}, in the case with zero element) as a linear basis, and with multiplicationand ⋆-operation coming from those of S.) Moreover, the unital ⋆-homomorphism C[S] →C*(G) extends by universality to C*(S), which is by definition the enveloping C*-algebraof C[S].We pause here to recall that S has a left regular representation ([1], p. 363); its extensionto C[S] is faithful ([15]), and this makes C*(S) to be a completion of C[S], and not of aproper quotient of it.

The envelopation of C[S] is done after all the unital ⋆-representationsof C[S] on Hilbert spaces, which correspond to ⋆-representations by partial isometries of S,and are all automatically bounded with respect to the l1-norm. (See [3] for more details.

)Hence the Lemma 6.1 can be rephrased:6.2 Proposition There exists a unital ⋆-homomorphism Ψ : C*(S) →C*(G), uniquelydetermined byΨ(α) = χA(α),α ∈S(6.2)(with A(α) as in (6.1), and χA(α) ∈Cc(G) its characteristic function).Let us next denote (as in the Example 1.4.1o above) by S(o) the subsemigroup of idempo-tents of S. Then clos sp{γ | γ ∈S(o)} ⊆C*(S) is, clearly, a unital Abelian C*-subalgebra,which will be denoted by C*(S(o)). (It is not difficult to show that, actually, this really22

is canonically isomorphic to the C*-algebra of the inverse semigroup S(o).) On the otherhand we denote (following [12], Section II.4) by C∗(G(o)) the unital Abelian C*-subalgebra{f ∈Cc(G) | supp f ⊆G(o) = {e}×Ω} of C*(G).

It is clear that Ψ of (6.2) induces a unital⋆-homomorphism Ψ(o) : C*(S(o)) →C*(G(o)).We have the following:6.3 Theorem In the context of Proposition 6.2, assume in addition that S is countableand that Ω(the space of the action) is second countable. Then:1o Ψ is onto if and only if Ψ(o) is so.2o Ψ is an isomorphism if and only if Ψ(0) is so.Proof If Ψ is onto, then Ψ(o) is onto.

Indeed, it is known ([12], Proposition II.4.8, wherewe compose on the right with the canonical surjection C*(G) →C*red(G) ) that there existsa conditional expectation P : C*(G) →C*(G(o)), such that P(f) = χG(o)f, f ∈Cc(G). Forα ∈S \ S(o) it is clear that P(Ψ(α)) = 0, and this immediately implies Ran(P ◦Ψ) ⊆Ran(Ψ(o)); but P ◦Ψ is onto, since P and Ψ are so, hence Ran(Ψ(o)) = C*(G(o)).It is obvious that Ψ faithful ⇒Ψ(o) faithful, so only the parts “⇐” of 1o and 2o aboveremain to be discussed.

We shall use the following6.3.1 Lemma Let x be in M (the index set for the maximal elements of S) and considerthe compact open G-set A(βx) = {x} × Dom(Φ(βx)) of G. Then Ψ induces a contractivelinear mapΨx : clos sp{α ∈S | α ≤βx} →{f ∈Cc(G) | supp f ⊆A(βx)},(6.3)where the latter set is a closed linear subspace of C*(G). If Ψ(o) is onto, then Ψx is onto; ifΨ(o) is an isomorphism, then Ψx is an isomorphism.Proof of Lemma 6.3.1 If C is a C*-algebra , Co ⊆C is a C*-subalgebra, and w ∈C is apartial isometry such that w∗w ∈Co, then wCo is a closed linear subspace of C (for instancebecause it can be written as {c ∈C | ww∗c = c, w∗c ∈Co}).

Assume in addition that wealso have Bo ⊆B C*-algebras, v ∈B partial isometry with v∗v ∈Bo, and Ψ : B →C ⋆-homomorphism such that Ψ(Bo) ⊆Co, Ψ(v) = w. Then: (a) Ψ(vBo) ⊆wCo; (b) Ψ(Bo) = Co⇒Ψ(vBo) = wCo; (c) Ψ|Bo faithful ⇒Ψ|vBo faithful. Indeed, (a),(b) are clear, while (c)23

comes out from the norm evaluation:||Ψ(vb)||2 = ||Ψb∗(v∗v)b||(⋆)= ||b∗(v∗v)b|| = ||vb||2,b ∈Bo,(where at (⋆) we used the faithfulness of Ψ|Bo).In our context, we have to put B = C*(S), Bo = C*(S(o)), v = βx and C = C*(G),Co = C*(G(o)), w = χA(βx) (Ψ is Ψ).We return to the proof of the Theorem. If Ψ(o) is onto, then from Lemma 6.3.1 and thedirect sum decomposition of (2.4) it is clear that RanΨ ⊇Cc(G), which is dense in C*(G);hence Ψ is onto.If Ψ(o) is an isomorphism, then (by the Lemma) every Ψx of (6.3) is a linear isomorphism.The family of linear maps (Ψ−1x )x∈M defines (due to the direct sum decomposition (2.4)) alinear map ρ : Cc(G) →C*(S).

This is easily checked to be a unital ⋆-homomorphism ofCc(G) (one uses the corresponding properties of Ψ). Now, due to the separability hypothesismade in the Theorem, the groupoid G is second countable, and we can (exactly as we didin the proofs of the Theorems 4.1 and 5.1) extend ρ to a unital C*-algebra homomorphismeρ : C*(G) →C*(S).

eρ clearly is an inverse for Ψ, which is hence an isomorphism. QEDThe homomorphism Ψ(o) of Theorem 6.3 has a natural interpretation as a map betweencompact spaces.

Indeed, let Z denote the spectrum of the (unital and Abelian) C*-algebraC*(S(o)); this is clearly homomorphic (and will be henceforth identified) to the space ofmultiplicative functions ζ : S(o) →{0, 1}, such that ζ(ǫ) = 1, ζ(θ) = 0 (where ǫ is theunit of S, and θ its zero element - if it exists). On the other hand, the convolution of thefunctions in C*(G(o)) coincides with their pointwise product, which makes clear that thespectrum of C*(G(o)) is canonically identified to Ω.Every ω ∈Ωgives a character ζω ∈Z, determined by ζω(γ) = 1, if ω ∈Dom(Φ(γ)),and ζω(γ) = 0 otherwise (γ ∈S(o)).

The mapψ(o) : Ω→Z,ψ(o)(ω) = ζω(6.4)is continuous, because Dom(Φ(γ)) is open and compact for every γ ∈S(o). It is an im-mediate verification (left to the reader) that ψ(o) is the map between character spacescorresponding to Ψ(o) : C*(S(o)) →C*(G(o)) of Theorem 6.3.

Hence we get:6.4 Corollary In the context of Theorem 6.3, the homomorphism Ψ : C∗(S) →C∗(G)is onto if and only if ψ(o) : Ω→Z of (6.4) is one-to one, i.e. if and only if the subsets24

(Dom(Φ(α)))α∈S separate the points of Ω. Moreover, Ψ is an isomorphism if and only ifψ(o) is bijective.6.5 Example Let us see how the Corollary 6.4 applies in the context of Toeplitz inversesemigroups.

Consider G, P, (βx)x∈G, SG,P as in Example 1.3, and let Φ : SG,P →IΩbethe action defined in Proposition 3.3. From equation (3.3) it is clear that Dom(Φ(α)) isopen and compact in Ω, for every α ∈SG,P, hence the considerations made in this sectioncan be applied.

Moreover, the family of subsets Dom(Φ(α))α∈SG,P separates the points ofΩ. Indeed, for A1 ̸= A2 in Ωwe can take an x ∈(A1 \ A2) ∪(A2 \ A1) ⊆PP −1 and it isobvious from equation (3.4) in Section 3 that Dom(Φ(βx)) will separate A1 from A2.Hence, we can construct the natural homomorphism Ψ : C*(SG,P) →C*(G), determinedby the equation (6.2), and moreover, Ψ is always surjective.From Corollary 6.4 it also comes out that Ψ will be an isomorphism if and only if thenatural map ψ(0) from Ωto the space Z of characters of C*(S(o)G,P ) is onto.

Concerning this,we mention without proof the following facts:(a) Z can be naturally identified to a subspace of {0, 1}G, in such a way that ψ(o) : Ω→Zbecomes an inclusion. This comes from the fact that S(o)G,P can be shown to be generatedby the family (β∗xβx)x∈P P −1; hence a multiplicative function on S(o)G,P is determined by itsvalues on this family, and we get an embedding τ : Z →{0, 1}G, defined by:τ(ζ) = {x ∈PP −1 | ζ(βxβ∗x) = 1}.

(6.5)It turns out that Ωψ(o)→Zτ→{0, 1}G is the inclusion of Ωinto {0, 1}G.(b) It is well-known that if E is a unital semilattice, then for every γ ∈E which isnot zero element we have a character ζγ of C*(E), determined by: ζγ(γ′) = 1, if γ′ ≥γ,ζγ(γ′) = 0, if γ′ ̸≥γ (γ′ ∈E); moreover, the family of the characters (ζγ)γ is dense inthe spectrum of C*(E). This offers the possibility of writing explicitly a dense subset ofτ(Z), i.e.

of Z identified inside {0, 1}G as in the preceding paragraph. The dense subset is{Bx1,...,xn | n ≥1, x1, .

. .

, xn ∈PP −1}, whereBx1,...,xn= {x ∈PP −1 | βxβ∗x ≥(β∗x1βx1) · · · (β∗xnβxn)}= {x ∈PP −1 | xP ⊇P ∩x−11 P ∩. .

. ∩x−1n P}.

(6.6)(c) Let us denote by “≺” the left-invariant partial pre-order determined by P on G,i.e.x ≺ydef⇔x−1y ∈P, for x, y ∈G.We shall call (G, P) “quasi-lattice ordered” ifP ∩P −1 = {e} and if:25

- every x ∈G having upper bounds in P ( equivalently, x ∈PP −1) has a least upperbound in P, denoted by σ(x);- every s, t ∈P having common upper bounds also have a least common upper bound,denoted by σ(s, t).The class of partially left-ordered groups satisfying these conditions contains the totallyleft-ordered groups, and is closed under direct products, semidirect products by order-preserving automorphisms, and free products (see [9]), Example 2.3).For (G, P) quasi-lattice ordered and such that, say, PP −1 ̸= G, the Toeplitz inversesemigroup SG,P is isomorphic to (P × P) ∪{θ}, with multiplication and ∗-operation:(s, t)(u, v) =((st−1σ(t, u), vu−1σ(t, u)),if t, u have common upper boundsθ,otherwiseθ(s, t) = (s, t)θ = θ = θ∗= θ2(s, t)∗= (t, s)(6.7)Indeed, (s, t) →βsβ∗t , θ →θ, is easily seen to define an isomorphism between (P × P) ∪{θ}and SG,P. (In the case when PP −1 = G, we have a similar isomorphism P × P →SG,P.

)If (G, P) is quasi-lattice ordered then it is immediately verified, using the particular formof SG,P, that the sets in (6.6) belong indeed to Ω. Hence in this case we have a canonicalisomorphism C*(SG,P) ≃C*(G).

(d) In general, the canonical homomorphism of C*(SG,P ) onto C*(G) is not faithful, asit can be seen on very simple examples which don’t have lattice properties. For instancefor G = Z2 and P = {(t1t2) ∈Z2 | 0 ≤t2 ≤2t1}, the identification of Z inside {0, 1}Gcontains elements which are not in Ω, and which are essentially produced by intersectionsof lines {t2 = a} ∩{2t1 −t2 = b} with a, b ∈N of different parity.6.6 Remark Let S be an eF-inverse semigroup.There always exists a “canonical”action of S which can be considered; namely, S acts by conjugation on the spectrum ofC*(S(o)), with S(o) the subsemigroup of idempotents of S (see [10], Section 3).

Indeed, S isfirst seen to act by conjugation on C*(S(o)) (this being an action by not necessarily unital⋆-endomorphisms); passing to the space Z of characters of C*(S(o)), we obtain an actionΦ : S →IZ described as follows:Dom(Φ(α)) = {ζ ∈Z | ζ(α∗α) = 1},α ∈SRan(Φ(α)) = {ζ ∈Z | ζ(αα∗) = 1},α ∈S(Φ(α))(ζ)(X) = ζ(α∗Xα)α ∈S, ζ ∈Dom(Φ(α)), X ∈C*(S(o)). (6.8)It is clear that Dom(Φ(α)) is an open and compact subset of Z, for every α ∈S, hencethe considerations made in this section can be applied.

Moreover, the map ψ(0) discussed26

in Corollary 6.4 clearly is, in this case, the identity map of Z. Hence if S is countable,Corollary 6.4 gives that the associated groupoid G has C*(G) ≃C*(S).27

References[1] B.A. Barnes.

Representations of the l1-algebra of an inverse semigroup, Trans. Amer.Math.

Soc. 218(1976), 361-396.

[2] J. Cuntz, W. Krieger. A class of C*-algebras and topological Markov chains, InventionesMath.

56(1980), 251-268. [3] J. Duncan, A.L.T.

Paterson. C*-algebras of inverse semigroups, Proc.

of the EdinburghMath. Soc.

28(1985), 41-58. [4] R. Exel.

Circle actions on C*-algebras, partial automorphisms and a generalizedPimsner-Voiculescu exact sequence, preprint. [5] R. Exel.

Approximately finite C*-algebras and partial automorphisms, preprint. [6] A. Kumjian.

On localizations and simple C*-algebras, Pacific J. Math.

112(1984), 141-192. [7] P. Muhly, J. Renault.

C*-algebras of multivariable Wiener-Hopf operators, Trans.Amer. Math.

Soc. 274(1982), 1-44.

[8] A. Nica. Some remarks on the groupoid approach to Wiener-Hopf operators, J. Oper-ator Theory 18(1987), 163-198.

[9] A. Nica. C*-algebras generated by isometries and Wiener-Hopf operators, PreprintINCREST No.33/1989, to appear in the Journal of Operator Theory.

[10] A.L.T. Paterson.

Inverse semigroups, groupoids, and a problem of J. Renault, preprint. [11] M. Petrich.

Inverse semigroups, John Wiley, 1984. [12] J. Renault.

A groupoid approach to C*-algebras, Lecture Notes in Math. 793, SpringerVerlag 1980.

[13] J. Renault. Representation des produits croises d’algebres de groupoides, J. OperatorTheory 18(1987), 67-97.

[14] S. Stratila, D. Voiculescu. Representations of AF-algebras and of the group U(∞),Lecture Notes in Math.

486, Springer Verlag 1975.28

[15] J.R. Wordingham. The left regular ⋆-representation of an inverse semigroup, Proc.Amer.

Math. Soc.

86(1982), 55-58.29

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