A correspondence between inverse subsemigroups, open wide subgroupoids and Cartan intermediate C*-subalgebras
AA CORRESPONDENCE BETWEEN INVERSESUBSEMIGROUPS, OPEN WIDE SUBGROUPOIDS ANDCARTAN INTERMEDIATE C*-SUBALGEBRAS
FUYUTA KOMURA
Abstract.
For a given inverse semigroup action on a topological space,one can associate an ´etale groupoid. We prove that there exists a corre-spondence between the certain subsemigroups and the open wide sub-groupoids in case that the action is strongly tight. Combining with therecent result of Brown et. al, we obtain a correspondence between thecertain subsemigroups of an inverse semigroup and the Cartan interme-diate subalgebras of a groupoid C*-algebra. Introduction
Given an action of an inverse semigroup on a topological space, one canassociate an ´etale groupoid by taking a germ. For a given ´etale groupoid, wecan construct groupoid C*-algebras, which are initiated by Renault [9]. Itis a natural task to investigate the relation among them and actually manyresearchers have been doing this. In this paper, we establish a correspon-dence between the set of certain subsemigroups and the set of wide openwide subgroupoids of the associated groupoids. We consider inverse semi-groups acting on topological spaces in the “strongly tight” way (see Defini-tion 2.1.1). Our main theorem, Theorem 2.1.10, states that wide open sub-groupoids of associated groupoids with strongly tight actions corresponds tocertain subsemigroups of the inverse semigroups. Combining with the workin [1], we obtain a correspondence between Cartan intermediate subalgebrasin groupoid C*-algebras and certain subsemigroups of inverse semigroups.As an application, we compute all Cartan intermediate subalgebras of theCuntz algebras which contains the fixed point algebras.This paper is organized as follows. Section 1 is devoted for preliminaries.In Section 2, we investigate open subgroupoids of ´etale groupoids associated
Department of Mathematics, Faculty of Science and Technology, Keio Uni-versity, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, JapanMathematical Science Team Center for Advanced Intelligence Project(AIP) RIKENPhone: +81-45-566-1641+42706Fax: +81-45-566-1642
E-mail address : [email protected] .1991 Mathematics Subject Classification. a r X i v : . [ m a t h . OA ] J u l to strongly tight actions. Then we establish a correspondence between openwide subgroupoids and certain subsemigroups (Theorem 2.1.10).In Section 3, we give applications of our correspondence. The first applica-tion is regarding with inverse semigroups which consist of compact bisectionsof ´etale groupoids. We show that a class of open wide subgroupoids of anample groupoid is described by an inverse semigroup of compact bisections(Corollary 3.1.3). As the second application, we study certain subsemi-groups of the polycyclic monoids. This study is applied to the computationof Cartan intermediate subalgebras between the Cuntz algebras and thefixed point algebras.In Section 4, we summarize the relation between Cartan intermediatesubalgebras of C*-algebras and certain subsemigroups of inverse semigroups.Then we compute Cartan intermediate subalgebras of the Cuntz algebraswhich contains the fixed point algebras.In Section 5, we mention the relation between strongly tight actions andtight groupoids. We give a characterization of a tight groupoid with thecompact unit space in Corollary 5.2.4. Acknowledgement.
The author would like to thank Prof. Takeshi Katsurafor his support and encouragement. This work was supported by JSPSKAKENHI 20J10088. 1.
Preliminaries
Inverse semigroups.
We recall the basic notions about inverse semi-groups. See [4] or [8] for more details. An inverse semigroup S is a semi-group where for every s ∈ S there exists a unique s ∗ ∈ S such that s = ss ∗ s and s ∗ = s ∗ ss ∗ . We denote the set of all idempotents in S by E ( S ) ·· = { e ∈ S | e = 2 } . It is known that E ( S ) is a commutative subsemi-group of S . An inverse semigroup which consists of idempotents is calleda (meet) semilattice. A zero element is a unique element 0 ∈ S such that0 s = s s ∈ S . A unit is a unique element 1 ∈ S suchthat 1 s = s s holds for all s ∈ S . In this paper, we assume thatevery inverse semigroup always has a zero element , although it doesnot necessarily have a unit. An inverse semigroup with a unit is called aninverse monoid. By a subsemigroup of S , we mean a subset of S which isclosed under the product and inverse of S . For s, t ∈ S , we write s ≤ t if ts ∗ s = s holds. Then this defines a partial order on S . Note that e ≤ f holds if and only if ef = e holds for e, f ∈ E ( S ). A pair s, t ∈ S is said to becompatible if s ∗ t, st ∗ ∈ E ( S ) holds. Notice that s, t are compatible if thereexists u ∈ S such that s, t ≤ u . A subsemigroup of an inverse semigroup S is said to be wide if it contains E ( S ). A subset I ⊂ E ( S ) is called an idealif e ∈ I and f ≤ e implies f ∈ I . A subset C ⊂ I of an ideal I ⊂ E ( S ) iscalled a cover if for every e ∈ I \ { } there exists c ∈ C such that ec (cid:54) = 0.For a topological space X , we denote by I X the set of all homeomorphismsbetween open sets in X . Then I X is an inverse semigroup with respect to the product defined by the composition of maps. For f, g ∈ I X , note that f ≤ g holds if and only if dom f ⊂ dom g and f ( x ) = g ( x ) hold for all x ∈ dom f .1.2. ´Etale groupoids. We recall the basic notions on ´etale groupoids. See[11] and [8] for more details.A groupoid is a set G together with a distinguished subset G (0) ⊂ G ,source and range maps d, r : G → G (0) and a multiplication G (2) ·· = { ( α, β ) ∈ G × G | d ( α ) = r ( β ) } (cid:51) ( α, β ) (cid:55)→ αβ ∈ G such that(1) for all x ∈ G (0) , d ( x ) = x and r ( x ) = x hold,(2) for all α ∈ G , αd ( α ) = r ( α ) α = α holds,(3) for all ( α, β ) ∈ G (2) , d ( αβ ) = d ( β ) and r ( αβ ) = r ( α ) hold,(4) if ( α, β ) , ( β, γ ) ∈ G (2) , we have ( αβ ) γ = α ( βγ ),(5) every γ ∈ G , there exists γ (cid:48) ∈ G which satisfies ( γ (cid:48) , γ ) , ( γ, γ (cid:48) ) ∈ G (2) and d ( γ ) = γ (cid:48) γ and r ( γ ) = γγ (cid:48) .Since the element γ (cid:48) in (5) is uniquely determined by γ , γ (cid:48) is called theinverse of γ and denoted by γ − . We call G (0) the unit space of G . Asubgroupoid of G is a subset of G which is closed under the inversion andmultiplication. A subgroupoid of G is said to be wide if it contains G (0) .A topological groupoid is a groupoid equipped with a topology where themultiplication and the inverse are continuous. A topological groupoid is saidto be ´etale if the source map is a local homeomorphism. Note that the rangemap of an ´etale groupoid is also a local homeomorphism. An ´etale groupoidis said to be ample if it has an open basis which consists of compact sets.In this paper, we mainly treat ample groupoids.A topological groupoid G is said to be topologically principal if the set { x ∈ G (0) | G ( x ) = { x }} is dense in G (0) , where G ( x ) is the isotropy group at x ∈ G (0) : G ( x ) ·· = { α ∈ G | d ( α ) = r ( α ) = x } . ´Etale groupoids associated to inverse semigroup actions. An´etale groupoid arises from an action of an inverse semigroup to a topolog-ical space. We recall how to construct an ´etale groupoid from an inversesemigroup action. We begin with the definition of an inverse semigroupaction.Let X be a topological space. Recall that I X is an inverse semigroup ofhomeomorphisms between open sets in X . An action α : S (cid:121) X is a semi-group homomorphism S (cid:51) s (cid:55)→ α s ∈ I X . In this paper, we always assumethat every action α satisfies (cid:83) e ∈ E ( S ) dom( α e ) = X and dom( α ) = ∅ . For e ∈ E ( S ), we denote the domain of α e by D αe . Then α s is a homeomorphismfrom D αs ∗ s to D αss ∗ . We often omit α of D αe if there is no chance to confuse. For an action α : S (cid:121) X , we associate an ´etale groupoid S (cid:110) α X as thefollowing. First we put the set S ∗ X ·· = { ( s, x ) ∈ S × X | x ∈ D αs ∗ s } . Thenwe define an equivalence relation ∼ on S ∗ X by declaring that ( s, x ) ∼ ( t, y )holds if x = y and there exists e ∈ E ( S ) such that x ∈ D αe and se = te. Set S (cid:110) α X ·· = S ∗ X/ ∼ and denote the equivalence class of ( s, x ) ∈ S ∗ X by[ s, x ]. The unit space of S (cid:110) α X is X , where X is identified with the subsetof S (cid:110) α X via the injection X (cid:51) x (cid:55)→ [ e, x ] ∈ S (cid:110) α X, x ∈ D αe . The source map and range maps are defined by d ([ s, x ]) = x, r ([ s, x ]) = α s ( x )for [ s, x ] ∈ S (cid:110) α X . The product of [ s, α t ( x )] , [ t, x ] ∈ S (cid:110) α X is [ st, x ]. Theinverse should be [ s, x ] − = [ s ∗ , α s ( x )]. Then S (cid:110) α X is a groupoid in theseoperations. For s ∈ S and an open set U ⊂ D αs ∗ s , define[ s, U ] ·· = { [ s, x ] ∈ S (cid:110) α X | x ∈ U } . These sets form an open basis of S (cid:110) α X . In these structures, S (cid:110) α X is an´etale groupoid.2. Correspondence between subsemigroups and subgroupoids
In this section, we consider strong tight actions of inverse semigroups(Definition 2.1.1). Then we establish a correspondence between certain sub-semigroups of an inverse semigroup and open wide subgroupoids of an ´etalegroupoid associated to a strongly tight action (Theorem 2.1.10). Then weobserve a condition for an open wide subgroupoid to be closed in terms ofan inverse semigroup.2.1.
Correspondence between subsemigroups and subgroupoids.
Webegin with the definition of a strongly tight action.
Definition 2.1.1.
Let S be an inverse semigroup and X be a locally com-pact Hausdorff space. An action α : S (cid:121) X is said to be ample if D αe ⊂ X isa compact set for all e ∈ E ( S ). We say that an action α : S (cid:121) X is stronglytight if { D αe } e ∈ E ( S ) is a basis of X .We remark that if there exists a strongly tight action α : S (cid:121) X , then X is totally disconnected and S (cid:110) α X is an ample groupoid.Strongly tight actions are related with the actions on tight spectrums ofinverse semigroups, which are investigated in [3]. We will see a relationbetween strongly tight actions and tight groupoids in Section 5.We construct subsemigroups from wide groupoids. Definition 2.1.2.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. Put G ·· = S (cid:110) α X . For a widesubgroupoid H ⊂ G , we define T H ·· = { s ∈ S | [ s, D s ∗ s ] ⊂ H } . Proposition 2.1.3.
In the above notations, T H is a wide subsemigroup of S . Proof . For e ∈ E ( S ), [ e, D e ] ∈ G (0) ⊂ H holds. Hence T H contains E ( S ).Next we show that T H is an subsemigroup of S . We show st ∈ T H for s, t ∈ T H . For x ∈ D ( st ) ∗ st , it follows that [ s, α t ( x )] , [ t, x ] ∈ H from s, t ∈ T H .Thus we obtain [ st, x ] = [ s, α t ( x )][ t, x ] ∈ H. Therefore we have [ st, D ( st ) ∗ st ] ⊂ H and st ∈ T H .It is clear that T H is closed under the inverse. Hence T H is a wide sub-semigroup of S . (cid:3) We define a class of subsemigroups which corresponds to open wide sub-groupoids (c.f. Theorem 2.1.10).
Definition 2.1.4.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. A wide subsemigroup T ⊂ S is said to be α -join closed if T has the next property :‘For every s ∈ S , s belongs to T if and only if there exists a finite set F ⊂ E ( S ) such that sf ∈ T holds for all f ∈ F and D s ∗ s ⊂ (cid:83) f ∈ F D f holds.’ Remark 2.1.5.
The “only if” part in the previous definition always holdsfor all wide subsemigroups T . Proposition 2.1.6.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. For a wide subgroupoid H ⊂ S (cid:110) α X , a wide subsemigroup T H ⊂ S is α -join closed. Proof . Take s ∈ S and assume that there exists a finite set F ⊂ E ( S ) suchthat sf ∈ T H for all f ∈ F and D s ∗ s ⊂ (cid:83) f ∈ F D f . It suffices to show s ∈ T H .For x ∈ D s ∗ s , there exists f ∈ F with x ∈ D f . Since we have sf ∈ T H , itfollows [ s, x ] = [ sf, x ] ∈ H. Thus we obtain [ s, D s ∗ s ] ⊂ H and therefore s ∈ T H . (cid:3) The proof of the next proposition is left to the reader.
Proposition 2.1.7.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. For a wide subsemigroup T ⊂ S , the map T (cid:110) α X (cid:51) [ t, x ] (cid:55)→ [ t, x ] ∈ S (cid:110) α X is an open map and an isomorphism onto its image. Via the map in the previous proposition, T (cid:110) α X is identified with thewide open subgroupoid of S (cid:110) α X . Lemma 2.1.8.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. For a wide subsemigroup T ⊂ S , T T (cid:110) α X ⊃ T holds. Moreover, if T is α -join closed, T T (cid:110) α X = T holds. Proof . The inclusion T T (cid:110) α X ⊃ T is clear. We assume that T is α -joinclosed and show T T (cid:110) α X ⊂ T . Take s ∈ T T (cid:110) α X and fix x ∈ D s ∗ s . Since wehave [ s, x ] ∈ T (cid:110) α X , there exists e x ∈ E ( S ) such that se x ∈ T and x ∈ D e x .Since we assume that D s ∗ s is compact, there exists a finite set P ⊂ D s ∗ s with D t ∗ t ⊂ (cid:83) x ∈ P D e x . Using the condition that T is α -join closed, we obtain t ∈ T . Now we have shown T T (cid:110) α X ⊂ T . (cid:3) Lemma 2.1.9.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an ample action. Put G ·· = S (cid:110) α X . Fora wide groupoid H ⊂ G , T H (cid:110) α X ⊂ H holds. Moreover, if H ⊂ G is openand α : S (cid:121) X is strongly tight, T H (cid:110) α X = H also holds. Proof . Assume that [ s, x ] ∈ T H (cid:110) α X . Then there exists t ∈ T H such that[ s, x ] = [ t, x ]. Now we have[ s, x ] = [ t, x ] ∈ [ t, D t ∗ t ] ⊂ H. Next we show the other inclusion T H (cid:110) α X ⊃ H under the assumptionthat α is strongly tight and H is open. Take [ s, x ] ∈ H . Since H is openand α is strongly tight, there exists e ∈ E ( S ) such that x ∈ D e ⊂ D s ∗ s and[ s, D e ] ⊂ H . One can see [ se, D ( se ) ∗ se ] ⊂ H , so we have se ∈ T H . Thereforeit follows [ s, x ] = [ se, x ] ∈ T H (cid:110) α X . (cid:3) The next theorem follows from Lemma 2.1.8 and Lemma 2.1.9.
Theorem 2.1.10.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. Assume that α : S (cid:121) X isstrongly tight. Let T denote the set of all wide α -join closed subsemigroupsof S . In addition, let H denote the set of all wide open subgroupoids of G .Then maps T (cid:51) T → T (cid:110) α X ∈ H and H (cid:51) H → T H ∈ T are inverse maps of each other.2.2. Closedness of subgroupoid.
We give conditions where T (cid:110) α X isclosed in S (cid:110) α X . Definition 2.2.1.
Let S be an inverse semigroup and T ⊂ S be a widesubsemigroup. For s ∈ S , we define J Ts ⊂ E ( S ) as J Ts ·· = { e ∈ E ( S ) | se ∈ T and e ≤ s ∗ s } . Proposition 2.2.2.
In the above notations, J Ts is an ideal of E ( S ). Proof . Assume e ∈ J Ts and f ≤ e . Then we have sf = sef ∈ T . Hencewe obtain f ∈ J Ts . (cid:3) We remark that J E ( S ) s = { e ∈ E ( S ) | e ≤ s } holds. This ideal appears in [3, Definition 3.11].Assume that an action α : S (cid:121) X is given. For an ideal J ⊂ E ( S ), wedefine D ( J ) ·· = (cid:83) e ∈J D e . The next lemma is a slight generalization of [3,Proposition 3.14]. Lemma 2.2.3.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. Assume that we are given awide subsemigroup T ⊂ S . Then the formula[ s, D ( J Ts )] = [ s, D s ∗ s ] ∩ ( T (cid:110) α X )holds for all s ∈ S . Proof . Take [ s, x ] ∈ [ s, D ( J Ts )]. Then there exists e ∈ J Ts with x ∈ D e .By the definition of J Ts , we have se ∈ T and e ≤ s ∗ s . Hence we obtain[ s, x ] = [ se, x ] ∈ [ s, D s ∗ s ] ∩ ( T (cid:110) α X ) . Now we have shown [ s, D ( J Ts )] ⊂ [ s, D s ∗ s ] ∩ ( T (cid:110) α X ). To show the reverseinclusion, take [ s, x ] ∈ [ s, D s ∗ s ] ∩ ( T (cid:110) α X ). Since [ s, x ] belongs to T (cid:110) α X ,there exists t ∈ T and f ∈ E ( S ) such that sf = tf and x ∈ D f hold. Sincewe have ss ∗ sf = sf = tf ∈ T , s ∗ sf belongs to J Ts . Since we also have x ∈ D s ∗ sf ⊂ J Ts , we obtain [ s, x ] ∈ [ s, D ( J Ts )]. (cid:3) Proposition 2.2.4.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be an action. Assume that we are given awide subsemigroup T ⊂ S . The following conditions are equivalent :(1) T (cid:110) α X is a closed subset of S (cid:110) α X ,(2) for every s ∈ S , D ( J Ts ) is a closed subset of D s ∗ s with respect tothe relative topology of D s ∗ s . Proof . First we show that (1) implies (2). By Lemma 2.2.3 and (1),[ s, D ( J Ts )] is a closed subset of [ s, D s ∗ s ]. Since the restriction of the domainmap d : [ s, D s ∗ s ] → D s ∗ s is a homeomorphism, d ([ s, D ( J Ts )]) = D ( J Ts ) isclosed in D s ∗ s . Next we show that (2) implies (1). It follows that [ s, D ( J Ts )]is a closed subset of [ s, D s ∗ s ] from the same argument in the above and (2).We have that [ s, D s ∗ s ] \ [ s, D ( J Ts )] is open in S (cid:110) α X since [ s, D s ∗ s ] is openin S (cid:110) α X and [ s, D ( J Ts )] is closed in [ s, D s ∗ s ]. One can see that the formula S (cid:110) α X \ T (cid:110) α X = (cid:91) s ∈ S ([ s, D s ∗ s ] \ [ s, D ( J Ts )])holds. Hence S (cid:110) α X \ T (cid:110) α X is open in S (cid:110) α X , which implies T (cid:110) α X isclosed in S (cid:110) α X . (cid:3) The next Lemma is essentially same as the [3, Proposition 3.7]. We givea proof for the reader’s convenience.
Lemma 2.2.5 (c.f. [3, Proposition 3.7]) . Let S be an inverse semigroup, X be a locally compact Hausdorff space and α : S (cid:121) X be a strongly tightaction. Assume that D e (cid:54) = ∅ holds for every e ∈ E ( S ) \ { } . For a ideal J ⊂ E ( S ) and a subset C ⊂ J , the followings are equivalent :(1) C is a cover of J ,(2) (cid:83) c ∈ C D c = D ( J ) holds. Proof . First we show (1) implies (2). The inclusion (cid:83) c ∈ C D c ⊂ D ( J )follows from C ⊂ J . We show the reverse inclusion. Take x ∈ D ( J ).Assume that x (cid:54)∈ D c holds for all c ∈ C . For each c ∈ C , there exists e c ∈ E ( S ) such that x ∈ D e c and D e c ∩ D c = ∅ since each D c is closed in X and { D e } e ∈ E ( S ) is a basis of X . Since D ce c = D c ∩ D e c = ∅ and we assume D e (cid:54) = ∅ for all e ∈ E ( S ) \ { } , we have ce c = 0. Putting p ·· = (cid:81) c ∈ C e c , wehave p ∈ J \ { } since J is ideal and x ∈ D p . However we also have cp = 0for each c ∈ C , which contradicts to the condition that C is a cover.Next we show (2) implies (1). Take e ∈ J \ { } . Then there exists c ∈ C such that D e ∩ D c (cid:54) = ∅ , which implies ec (cid:54) = 0. Hence C is a cover of J . (cid:3) Now we obtain the characterization about the closedness of open widesubgroupoids.
Theorem 2.2.6.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be a strongly tight action. Assume that D e (cid:54) = ∅ holds for every e ∈ E ( S ) \ { } . For a wide subsemigroup T ⊂ S , thefollowing conditions are equivalent :(1) T (cid:110) α X is closed in S (cid:110) α X ,(2) for every s ∈ S , D ( J Ts ) is relatively closed in D s ∗ s ,(3) for every s ∈ S , J Ts has a finite cover. Proof . Now it suffices to show that (2) and (3) are equivalent, since Propo-sition 2.2.4 states that (1) and (2) are equivalent. First we show that (2)implies (3). Since we assume that the action α is ample, D s ∗ s is compact.Then D ( J Ts ) is also compact by (2). Hence there exists a finite set C ⊂ J Ts such that (cid:83) c ∈ C D c = D ( J Ts ). By Lemma 2.2.5, C is a finite cover of J Ts .Next we show that (3) implies (2). Take s ∈ S and a finite cover C of J Ts .By Lemma 2.2.5 again, we have D ( J ) = (cid:83) c ∈ C D c . Hence we have D ( J ) iscompact and therefore closed in D s ∗ s since each D c is compact. (cid:3) Wide clopen subgroupoids arise from partial group homomorphisms. Weobserve this fact in the remainder of this subsection.Let S be an inverse semigroup and Γ be a group. Put S × ·· = S \ { } .A map σ : S × → Γ is called a partial homomorphism if σ ( st ) = σ ( s ) σ ( t )holds for any pair s, t ∈ S × with st (cid:54) = 0. A partial homomorphism gives usa suitable subsemigroup as follows. Proposition 2.2.7.
Let S be an inverse semigroup, Γ be a group and σ : S × → Γ be a partial homomorphism. Assume that we are given a locallycompact space X and an action α : S (cid:121) X where D e (cid:54) = ∅ holds for each e ∈ E ( S ) \ { } . Then the following statements hold :(1) ker σ ·· = σ − ( e ) ∪ { } is a α -join closed wide subsemigroup of S ,(2) ker σ (cid:110) α X is closed in S (cid:110) α X . Proof . First we show (1). One can see that ker σ is a wide subsemigroupof S in a straightforward way. We show ker σ is α -join closed. Take s ∈ S and assume that there exists a finite set F ⊂ E ( S ) such that sF ⊂ ker σ and D s ∗ s ⊂ (cid:83) f ∈ F D f . It suffices to show s ∈ ker σ . We may assume that s (cid:54) = 0. Then there exists f ∈ F such that D s ∗ s ∩ D f (cid:54) = ∅ , which implies sf (cid:54) = 0. Since we have sf ∈ ker σ , it follows σ ( s ) = σ ( s ) e = σ ( s ) σ ( f ) = σ ( sf ) = e. Hence s ∈ ker σ .Next we show (2). Although it is possible to apply Proposition 2.2.4, weshow (2) using a cocycle on a groupoid. We define the map c σ : S (cid:110) α X → Γby c σ ([ s, x ]) = σ ( s ) , [ s, x ] ∈ S (cid:110) α X. Then c σ is a continuous cocycle. One can see thatker σ (cid:110) α X = c − σ ( e )holds. Hence ker σ (cid:110) α X is closed in S (cid:110) α X . (cid:3) Applications and examples
Inverse semigroups of compact bisections.
Let G be an ample´etale groupoid. Recall that an open set U ⊂ G is called a bisection if therestrictions d | U and r | U are homeomorphisms onto the images. Let I ( G )denote the set of all compact bisections of G . For U, V ∈ I ( G ), their product U V is defined by
U V ·· = { αβ ∈ G | α ∈ U, β ∈ V, d ( α ) = r ( β ) } . Then
U V belongs to I ( G ). It is known that I ( G ) becomes an inverse semi-group. Note that the inverse of U ∈ I ( G ) is given by U − = { α − ∈ G | α ∈ U } . The order of I ( G ) as an inverse semigroup coincides with the order definedby inclusion. A pair U, V ∈ I ( G ) is said to be compatible if U − V and U V − belongs to E ( I ( G )). If U, V ∈ I ( G ) are compatible, U ∪ V is an elementof I ( G ). Note that U ∪ V is the least upper bound of { U, V } . Thus I ( G )admits joins of compatible pairs in I ( G ). A subsemigroup T ⊂ I ( G ) is saidto be join closed if all joins of compatible pair of T also belongs to T . A map from a groupoid to a group is called a cocycle if it preserves the products. For U ∈ I ( G ), we have a homeomorphism ρ U : d ( U ) → r ( U ) defined by ρ U ( d ( α )) = r ( α ) , α ∈ U. Then the map U (cid:55)→ ρ U defines an action ρ : I ( G ) (cid:121) G (0) . One can see that ρ is strongly tight. The following theorem is essentially same as [7, Theorem2.8]. Theorem 3.1.1 (c.f. [7, Theorem 2.8]) . Let G be an ample ´etale groupoid.Then G is isomorphic to I ( G ) (cid:110) ρ G (0) . Proof . For α ∈ G , there exists U α ∈ I ( G ) such that α ∈ U α since G isample. Then [ U α , d ( α )] ∈ I ( G ) (cid:110) ρ G (0) is independent of the choice of U α .Thus we obtain the mapΦ : G (cid:51) α (cid:55)→ [ U α , d ( α )] ∈ I ( G ) (cid:110) ρ G (0) . One can see that Φ is an isomorphism as a morphism between ´etale groupoids.Indeed, the map Ψ : I ( G ) (cid:110) ρ G (0) → G defined by Ψ([ U, x ]) = d − U ( x ) is theinverse map of Φ. (cid:3) Lemma 3.1.2.
Let G be an ample ´etale groupoid. Then a wide subsemi-group T ⊂ I ( G ) is ρ -join closed if and only if T is join closed. Proof . Assume that T ⊂ I ( G ) is join closed. Take U ∈ I ( G ) and thereexists a finite set F ⊂ E ( I ( G )) such that U F ∈ T and D ρU ∗ U ⊂ (cid:83) O ∈F D ρO .Observe that elements in U F are pairwisely compatible and (cid:87) O ∈F U O = U holds. Since T is join closed, U belongs to T .To show the converse, assume that T ⊂ I ( G ) is ρ -join closed. Let U, V ∈ T be compatible. Put F ·· = { U − U, V − V } ⊂ E ( I ( G )). Then one can see that( U ∪ V ) F = { U, V } ⊂ T hold. Since we havedom( ρ U ∪ V ) = d ( U ) ∪ d ( V ) = dom ρ U ∪ dom ρ V , we obtain U ∪ V ∈ T by the ρ -closedness of T . (cid:3) Theorem 2.1.10, Theorem 3.1.1 and Lemma 3.1.2 yield the next corollary.
Corollary 3.1.3.
Let G be an ample ´etale groupoid. Then there is a cor-respondence between the set of all open wide subgroupoids of G and the setof all wide join closed subsemigroups of I ( G ).3.2. Polycyclic monoids.
We apply Theorem 2.1.10 to the polycyclic monoids P n . Definition 3.2.1.
Let n ≥ P n is an inverse monoid defined by P n ·· = (cid:104) s , s , . . . , s n | s ∗ i s j = δ i,j (cid:105) . See [5] for details on the polycyclic monoids.Set Σ n ·· = { , , . . . , n } andΣ N n ·· = { ( x i ) ∞ i =1 | x i ∈ Σ n for all i ∈ N } . It follows that Σ N n is a compact Hausdorff space from Tychonoff’s theorem.We write a finite sequence on Σ n like µ = ( µ , µ , . . . , µ l ), where each µ j isan element of Σ n . Here, l ∈ N is called the length of µ , which we denoteby | µ | . The only element of length zero is denoted by ε , which is calledthe empty word. We denote the set of all finite sequence on Σ n by Σ ∗ n .For µ ∈ Σ ∗ n , we define a cylinder set C ( µ ) ⊂ Σ N n as the set of all infinitesequences which begin with µ : C ( µ ) ·· = { ( x i ) ∞ i =1 ∈ Σ N n | x i = µ i for all i = 1 , , . . . , | µ |} . We represent an element of C ( µ ) as µx with x ∈ Σ N n . Each C ( µ ) is a compactopen set of Σ N n and the family of all C ( µ ) is a basis of Σ N n . For µ ∈ Σ ∗ n , wedefine s µ ∈ P n as s µ ·· = s µ s µ · · · s µ | µ | . For the empty word ε ∈ Σ N n , we define s ε = 1. It is known that an elementof P n \ { } is represented as the form s µ s ∗ ν for unique µ, ν ∈ Σ ∗ n .Now we define an action β : P n (cid:121) Σ N n . For s µ s ∗ ν ∈ P n , define β s µ s ∗ ν : C ( µ ) → C ( ν ) by β s µ s ∗ ν ( νx ) = µx, x ∈ Σ N n . Then the map s µ s ∗ ν (cid:55)→ β s µ s ∗ ν defines an action β : P n (cid:121) Σ N n . Since thedomain of s µ s ∗ µ coincides with C ( µ ), the action β is strongly tight.For k, l ∈ N , we define P k,ln ·· = { s µ s ∗ ν ∈ P n | | µ | = k, | ν | = l } . Observe that M n ·· = (cid:91) k ∈ N P k,kn ∪ { } = { s µ s ∗ ν ∈ P n | | µ | = | ν |} ∪ { } is a wide subsemigroup of P n .We investigate β -join closed subsemigroups T ⊂ P n such that M n ⊂ T .For m ∈ N , define P mn ·· = (cid:91) k − l ∈ m Z P k,ln ∪ { } . Then one can see that P mn is a β -join closed subsemigroup which contains M n . Notice that P n = M n . Conversely, we obtain the following proposition. Proposition 3.2.2.
Assume that T (cid:40) P n is a β -join closed subsemigroupwhich contains M n . Then T = P mn holds for some m ∈ N .In order to prove this proposition, we prepare a few lemmas. The nextlemma follows from straightforward calculations. Lemma 3.2.3.
For i, j, k, l ∈ N , we have P i,jn P k,ln = (cid:40) P i + k − j,ln ∪ { } ( k ≥ j ) ,P i,j − k + ln ∪ { } ( k ≤ j ) . Lemma 3.2.4.
Let T ⊂ P n be a wide subsemigroup which contains M n .Then the following statements hold :(1) If s µ s ∗ ν ∈ T holds, then P | µ | , | ν | n ⊂ T holds.(2) P k,ln ⊂ T implies P l,kn ⊂ T .(3) P k,ln ⊂ T implies P k +1 ,l +1 n ⊂ T .Moreover, if T is β -join closed, then the following holds :(4) If P k,ln holds for k, l ∈ Z > , then P k − ,l − n ⊂ T holds. Proof . (1) Assume | µ | = | µ (cid:48) | and | ν | = | ν (cid:48) | hold for µ (cid:48) , ν (cid:48) ∈ Σ N n . Then wehave s µ (cid:48) s ∗ µ , s ν s ∗ ν (cid:48) ∈ M n ⊂ T . Since we assume s µ s ∗ ν ∈ T , it follows s µ (cid:48) s ν (cid:48) = s µ (cid:48) s ∗ µ s µ s ∗ ν s ν s ∗ ν (cid:48) ∈ T. Hence we have P | µ | , | ν | n ⊂ T .(2) is clear, so we show (3) next. Take s µ s ∗ ν ∈ P k,ln and x, y ∈ Σ n arbi-trarily. Then we have s µx s ∗ νx = s µ s ∗ ν s νx s ∗ νx ∈ T, where we use the fact s νx s ∗ νx ∈ M n ⊂ T . Using (1) in the above, we obtain P k +1 .l +1 n ⊂ T .Finally we show (4) under the assumption that T is β -join closed. Take s µ s ∗ ν ∈ P k − ,l − n . For each x ∈ Σ n , we have s µ s ∗ ν s νx s ∗ νx = s µx s ∗ νx ∈ T, since we assume P k,ln ⊂ T . Observe that D β ( s µ s ∗ ν ) ∗ s µ s ∗ ν = D βs ν s ∗ ν = (cid:91) x ∈ Σ n D βs νx s ∗ νx (= C ( ν )) . Since T is β -join closed, we have s µ s ∗ ν ∈ T . Hence we have shown P k − ,l − n ⊂ T . (cid:3) Proof of Proposition 3.2.2 . We may assume that M n (cid:40) T . We define m ·· = min {|| µ | − | ν || ∈ N | s µ s ∗ ν ∈ T \ M n } ( > . We show T = P mn . By the definition of m , there exists s µ s ∗ ν ∈ T suchthat || µ | − | ν || = m . Since T is closed under the inverse, we may assume | µ | − | ν | = m . Using (1) of Lemma 3.2.4, we have P | µ | , | ν | m ⊂ T . Applying(4) of Lemma 3.2.4 repeatedly, we obtain P m, n ⊂ T and it follows P ,mn ⊂ T from (2) of Lemma 3.2.4. Now one can see that P k,ln ⊂ T holds for k, l with k − l ∈ m Z . Hence we obtain P mn ⊂ T .Next we show T ⊂ P mn . Assume that there exists s µ s ∗ ν ∈ T such that s µ s ∗ ν (cid:54)∈ P mn . We may assume that | µ | > | ν | . Take a, b ∈ N such that | µ | − | ν | = am + b and 1 ≤ b ≤ m −
1. We have P | µ | , | ν | n ⊂ T by (1) of Lemma P am + b, n ⊂ T . Since wehave P m, n ⊂ T , it follows P ( a − m + b, n ⊂ P am + b, n P m, n ⊂ T, where we used Lemma 3.2.3. Repeating this argument inductively, we obtain P b, n ⊂ T . This contradicts to the minimality of m . Now we have shown T = P mn . (cid:3) By Theorem 2.1.10 and Proposition 3.2.2, an open proper intermediatesubgroupoid between P n (cid:110) α Σ N n and M n (cid:110) β Σ N n is given by the form P mn (cid:110) β Σ N n for some m ∈ N . Now we see P mn (cid:110) β Σ N n is closed. Observe that P n (cid:110) β Σ N n has a continuous cocycle c : P n (cid:110) β Σ N n → Z defined by c ([ s µ s ∗ ν , x ]) = | µ | − | ν | .Since we have P mn (cid:110) β Σ N n = c − ( m Z ), it follows that P mn (cid:110) β Σ N n is a closedsubset of P n (cid:110) β Σ N n . Hence we obtain the next proposition. Proposition 3.2.5.
Every open wide normal subgroupoid of P n (cid:110) β Σ N n whichcontains M n (cid:110) β Σ N m is closed.It follows from Corollary 5.2.4 that P n (cid:110) β Σ N n is isomorphic to the tightgroupoid of P n . See Section 5 for more details.4. Applications to the theory of C*-algebras
Analysis of Cartan intermediate subalgebras by using inversesemigroups.
In this section, we explain a correspondence between Cartanintermediate subalgebras and certain subsemigroups of an inverse semigroup.
Definition 4.1.1.
Let A be a C*-algebra. A commutative subalgebra D ⊂ A is called a Cartan subalgebra if the following conditions hold :(1) The inclusion D ⊂ A is nondegenerate (i.e. D contains an approxi-mate unit for A ).(2) The set of normalizers generates A , where n ∈ A is called a normal-izer if nDn ∗ ∪ n ∗ Dn ⊂ D holds.(3) There is a faithful conditional expectation E : A → D .(4) The commutant D (cid:48) coincides with D , where D (cid:48) ·· = (cid:84) d ∈ D { a ∈ A | da = ad } .We call ( A, D ) a Cartan pair. An intermediate subalgebra D ⊂ B ⊂ A iscalled a Cartan intermediate subalgebra if D is Cartan in B .Renault’s cerebrated theorem states that a Cartan pair arises from atwisted groupoid. We refer to [1] and [11] for twists of ´etale groupoids.A twisted groupoid over G is a topological groupoid Σ with the centralextension G (0) × T (cid:44) → Σ q (cid:16) G, where T is the circle group. In this paper, this twist is abbreviated to q : Σ → G . We denote the reduced C*-algebra of the twist q : Σ → G by C ∗ λ (Σ). Recall that C ∗ λ (Σ) contains C ( G (0) ) as a subalgebra. We denote the reduced C*-algebra of G by C ∗ λ ( G ), which is isomorphic to the reducedC*-algebra of the trivial twist G × T → G . Theorem 4.1.2 ([10, Theorem 5.9]) . Let (
A, D ) be a Cartan pair where A is separable. Then there exists a twist q : Σ → G such that A is isomorphicto C ∗ λ (Σ) via an isomorphism which maps D to C ( G (0) ), where G is secondcountable topologically principal locally compact Hausdorff ´etale groupoid.This twist q : Σ → G is unique up to isomorphism.From now on, we identify C ∗ λ (Σ) and C ( G (0) ) with A and D respectivelyfor a Cartan pair ( A, D ).Let q : Σ → G be a twist and H ⊂ G be a wide open subgroupoid. [1,Lemma 3.2] states that Σ H ·· = q − ( H ) naturally becomes a twist over H and there exists a natural inclusion C ∗ λ (Σ H ) ⊂ C ∗ λ (Σ). The authors in [1]showed this map H (cid:55)→ C ∗ λ (Σ H ) gives a certain correspondence as follows. Theorem 4.1.3 ([1, Theorem 3.3, Lemma 3.4]) . Let (
A, D ) be a Cartanpair with a separable A and q : Σ → G be an associated twist. Then theabove map H (cid:55)→ C ∗ λ (Σ H ) gives a one-to-one correspondence between theset of open wide subgroupoids of G and the set of Cartan intermediatesubalgebras D ⊂ B ⊂ A . In addition, there exists a conditional expectationfrom C ∗ λ (Σ) to C ∗ λ (Σ H ) if and only if H ⊂ G is closed.Combining Theorem 4.1.3 with Theorem 2.1.10, we obtain the next Corol-lary. Corollary 4.1.4.
Let (
A, D ) be a Cartan pair with separable A and q : Σ → G be an associated twist. Assume that G = S (cid:110) α X holds for some stronglytight action α : S (cid:121) X . Then there exists a one-to-one correspondencebetween the set of α -join closed wide subsemigroups of S and the set ofCartan intermediate subalgebras D ⊂ B ⊂ A . More precisely, the map T (cid:55)→ C ∗ λ (Σ T (cid:110) α X ) gives the above correspondence. Example 4.1.5.
We investigate certain subalgebras of the Cuntz algebrasby using the polycyclic monoids here. For n ∈ N with n ≥
2, the Cuntz alge-bra O n is the universal unital C*-algebra generated by isometries S , . . . , S n which satisfy Cuntz relation as follows : S ∗ i S j = δ i,j , n (cid:88) i =1 S i S ∗ i = 1 . For a finite sequence µ = ( µ , . . . , µ l ) on { , . . . , n } , we define S µ ·· = S µ S µ · · · S µ l . Then O n is the closure of the linear span of { S µ S ∗ ν } µ,ν , where µ and ν aretaken over the all finite sequences on { , . . . , n } . Let D n be the subalgebraof O n generated by { S µ S ∗ µ } µ , where µ is taken over the all finite sequenceson { , . . . , n } . We denote the gauge action by τ : T (cid:121) O n . Note that the gauge action satisfies τ z ( S i ) = zS i for all z ∈ T and i = 1 , , . . . , n . Wedenote the fixed point algebra of τ by O τn ·· = (cid:92) z ∈ T { x ∈ O n | τ z ( x ) = x } . Then O τn is the closure of the linear span of { S µ S ∗ ν ∈ O n | | µ | = | ν |} , where | µ | denotes the length of µ .The polycyclic monoids have strongly tight actions β : P n (cid:121) Σ N n , de-scribed in subsection 3.2. Put G n ·· = P n (cid:110) β Σ N n . Then G n is a topologicallyprincipal locally compact Hausdorff second countable ample groupoid. For s i ∈ P n , let χ [ s i ,D s ∗ i si ] denote the characteristic function on [ s i , D s ∗ i s i ] ⊂ G n .Then { χ [ s i ,D s ∗ i si ] } ni =1 are elements of C ∗ λ ( G n ) and generate C ∗ λ ( G n ). Since { χ [ s i ,D s ∗ i si ] } ni =1 satisfies the Cuntz relation, O n and C ∗ λ ( G n ) are isomorphicvia the unique isomorphism Φ : O n → C ∗ λ ( G n ) such that Φ( S i ) = χ [ s i ,D s ∗ i si ] holds for all i = 1 , . . . , n . One can see thatΦ( D n ) = C (Σ N n ) and Φ( O τn ) = C ∗ λ ( M n (cid:110) β Σ N n )hold. Define O mn ⊂ O n to be the subalgebra generated by { S µ S ∗ ν ∈ O n | | µ | − | ν | ∈ m Z } . One can see that Φ( O mn ) = C ∗ λ ( P mn (cid:110) β Σ N n )holds. Therefore it follows from Proposition 3.2.2 that a Cartan intermedi-ate subalgebra O τn ⊂ B ⊂ O n coincides with O mn for some m ∈ N . Moreover,every Cartan intermediate subalgebra between O τn and O n admits a condi-tional expectation from O n by Proposition 3.2.5 and Theorem 4.1.3.We note that O mn is isomorphic to O n m . Indeed, { S µ } | µ | = m generates O mn and satisfies the Cuntz relation.5. Relation between strongly tight actions and tightgroupoids
In this section, we observe that tight groupoids, which are investigated in[3], are related with strongly tight actions.5.1.
Tight groupoids.
First we recall the definition of tight groupoids.Refer to [2] or [3] for more details. Let S be an inverse semigroup. Acharacter on E ( S ) is a nonzero semigroup homomorphism from E ( S ) to { , } , where { , } is equipped with the usual multiplication. We denotethe set of all characters on E ( S ) by (cid:98) E ( S ). Letting (cid:98) E ( S ) be equipped withthe pointwise convergence topology, (cid:98) E ( S ) is a locally compact Hausdorffspace. For a ξ ∈ (cid:98) E ( S ), ξ − ( { } ) ⊂ E ( S ) is a proper filter in the followingsense : (1) ξ − ( { } ) does not contain 0,(2) if e and f belongs to ξ − ( { } ), then ef also belongs to ξ − ( { } ),(3) if e ∈ ξ − ( { } ) and f ≥ e hold, then f belongs to ξ − ( { } ).A character ξ ∈ (cid:98) E ( S ) is called an ultracharacter if ξ − ( { } ) is a maximalproper filter. A character ξ ∈ (cid:98) E ( S ) is an ultracharacter if and only if there isno character η ∈ (cid:98) E ( S ) such that ξ < η holds. The set of all ultracharacterson E ( S ) is denoted by (cid:98) E ∞ ( S ). The closure of (cid:98) E ∞ ( S ) in (cid:98) E ( S ) is denoted by (cid:98) E tight ( S ). An element in (cid:98) E tight ( S ) is called a tight character.We define the spectral action β : S (cid:121) (cid:98) E ( S ). For e ∈ E ( S ), put D βe ·· = { ξ ∈ (cid:98) E ( S ) | ξ ( e ) = 1 } . Note that D βe is a compact open set of (cid:98) E ( S ). For s ∈ S and ξ ∈ D βs ∗ s , define β s ( ξ ) ∈ D βss ∗ by β s ( ξ )( e ) ·· = ξ ( s ∗ es ) , e ∈ E ( S ) . Then β s : D βs ∗ s → D βss ∗ is a homeomorphism. The map s (cid:55)→ β s defines anaction β : S (cid:121) (cid:98) E ( S ). It is known that (cid:98) E ∞ ( S ) and (cid:98) E tight ( S ) are β -invariant(see [2, Proposition 12.11]). The restrictions of β to (cid:98) E ∞ ( S ) and (cid:98) E tight ( S )are denoted by θ ∞ : S (cid:121) (cid:98) E ∞ ( S ) and θ : S (cid:121) (cid:98) E tight ( S )respectively. The tight groupoid of S is defined as G tight ( S ) ·· = S (cid:110) θ (cid:98) E tight ( S ).5.2. Characterization of tight groupoids.
Strongly tight actions withnonempty domains are characterized as the following theorem.
Theorem 5.2.1.
Let S be an inverse semigroup, X be a locally compactHausdorff space and α : S (cid:121) X be a strongly tight action such that D e (cid:54) = ∅ holds for each e ∈ E ( S ) \ { } . Then there exists a homeomorphism X (cid:51) x (cid:55)→ ξ x ∈ (cid:98) E ∞ ( S )which induces an isomorphism S (cid:110) α X (cid:51) [ s, x ] (cid:55)→ [ s, ξ x ] ∈ S (cid:110) θ ∞ (cid:98) E ∞ ( S ) . Proof . This is a simple modification of [12, Proposition 5.5]. We give aproof for the reader’s convenience.For x ∈ X , we define ξ x ∈ (cid:98) E ( S ) by ξ x ( e ) ·· = (cid:40) x ∈ D αe ) , x (cid:54)∈ D αe ) . We show ξ x ∈ (cid:98) E ∞ ( S ). Assume that there exists η ∈ (cid:98) E ( S ) such that ξ x < η .Then there exists f ∈ E ( S ) such that ξ x ( f ) = 0 and η ( f ) = 1. Since weassume that { D αe } e ∈ E ( S ) is a basis of X , there exists e ∈ E ( S ) such that x ∈ D αe and D αe ∩ D αf = ∅ . By the assumption ξ x < η , we have η ( e ) = 1. By D αef = D αe ∩ D αf = ∅ , we have ef = 0 and therefore η ( ef ) = 0. Thiscontradicts to η ( ef ) = η ( e ) η ( f ) = 1. Hence ξ x is an ultracharacter.We define the map Φ : X (cid:51) x (cid:55)→ ξ x ∈ (cid:98) E ∞ ( S ). We show that Φ is ahomeomorphism. It is easy to show that Φ is continuous. To show that Φis injective, take x, y ∈ X with x (cid:54) = y . Since a family { D αe } e ∈ E ( S ) is a basisof X , there exists e ∈ E ( S ) such that x ∈ D αe and y (cid:54)∈ D αe . Then ξ x ( e ) = 1and ξ y ( e ) = 0. Therefore we have ξ x (cid:54) = ξ y and Φ is injective.Next, take ξ ∈ (cid:98) E ∞ ( S ) to show that Φ is surjective. Because a family { D αe | ξ ( e ) = 1 } has the finite intersection property, (cid:84) ξ ( e )=1 D αe is notempty. Take x ∈ (cid:84) ξ ( e )=1 D αe . Then we have ξ ≤ ξ x . By the maximality of ξ , we obtain ξ = ξ x . Therefore the map x (cid:55)→ ξ x is surjective.Now one can check that Φ( D αe ) = D βe holds. Using this, it follows that Φis a homeomorphism.It is straightforward to check that there exists a (unique) isomorphismwhich maps [ s, x ] ∈ S (cid:110) α X to [ s, ξ x ] ∈ S (cid:110) θ ∞ (cid:98) E ∞ ( S ). (cid:3) Remark 5.2.2.
It seems difficult to drop the assumption that D e (cid:54) = ∅ for e ∈ E ( S ) \ { } . Define matrices p = (cid:18) (cid:19) , q = (cid:18) (cid:19) . Then E ·· = { , p, q, } is a semilattice with respect to the usual multiplica-tion. Let X = { x } be a singleton. Define an action α : E (cid:121) X by declaring D = X and D p = D q = D = ∅ . Then α is strongly tight, although (cid:98) E ∞ is not homeomorphic to X . Note that ξ x , which is defined in the proof ofTheorem 5.2.1, is not an ultracharacter. Therefore it seems difficult to finda natural map between X and (cid:98) E ∞ .The author in [6] showed the following theorem. Theorem 5.2.3 ([6, Theorem 2.5]) . Let E be a semilattice with zero andunit elements. Then (cid:98) E ∞ = (cid:98) E tight holds if and only if (cid:98) E ∞ is compact.Theorem 5.2.1 and Theorem 5.2.3 yield the following characterization oftight groupoids. Corollary 5.2.4.
Let S be an inverse semigroup. Consider the followingconditions.(1) S has a strongly tight action to a compact Hausdorff set X .(2) (cid:98) E ( S ) ∞ is compact,(3) (cid:98) E ( S ) tight = (cid:98) E ( S ) ∞ .Then (1) ⇔ (2) and (2) ⇒ (3) hold. If S has a unit element, (3) ⇒ (2) alsoholds. Moreover, if (1) holds, then S (cid:110) X is isomorphic to G tight ( S ). Remark 5.2.5.
The implication (3) ⇒ (2) in Corollary 5.2.4 dose notnecessarily hold in general. Let E be a semilattice generated by 0 and { p i } i ∈ N with the relation p i p j = (cid:40) p i ( i = j ) , i (cid:54) = j ) . Then (cid:98) E ∞ = (cid:98) E tight holds, although (cid:98) E ∞ is not compact. Indeed (cid:98) E ∞ is home-omorphic to N . Remark 5.2.6.
There exists a semilattice E such that (cid:98) E ∞ is a locallycompact although (cid:98) E ∞ (cid:40) (cid:98) E tight holds. Let E be the semilattice in Remark5.2.5. Put E ·· = E ∪{ } . Then (cid:99) E ∞ is locally compact. In addition we have (cid:99) E ∞ (cid:40) (cid:99) E . Indeed, (cid:99) E ∞ and (cid:99) E are isomorphic to N and N ∪ {∞} respectively. Therefore we can not relax the condition (2) in Corollary 5.2.4. References [1] J. H. Brown, R. Exel, A. H. Fuller, D. R. Pitts, and S. A. Reznikoff. IntermediateC*-algebras of Cartan embeddings. arXiv: 1912.03686 .[2] R. Exel. Inverse semigroups and combinatorial C*-algebras.
Bulletin of the BrazilianMathematical Society, New Series , 39, 04 2007.[3] R. Exel and E. Pardo. The tight groupoid of an inverse semigroup.
Semigroup Forum ,92(1):274–303, 2016.[4] M. V. Lawson.
Inverse Semigroups: The Theory of Partial Symmetries . World Sci-entific, 1998.[5] M. V. Lawson. The Polycyclic Monoids P n and the Thompson Groups V n, . Commu-nications in Algebra , 35(12):4068–4087, 2007.[6] M. V. Lawson. Compactable semilattices.
Semigroup Forum , 81(1):187–199, 2010.[7] D. Matsnev and P. Resende. ´Etale groupoids as germ groupoids and their base ex-tensions.
Proceedings of the Edinburgh Mathematical Society , 53(3):765–785, 2010.[8] A. Paterson.
Groupoids, Inverse Semigroups, and their Operator Algebras . Progressin Mathematics. Birkh¨auser Boston, 2012.[9] J. Renault.
A Groupoid Approach to C*-Algebras . Lecture Notes in Mathematics.Springer-Verlag, 1980.[10] J. Renault. Cartan subalgebras in C*-algebras.
Irish Math. Soc. Bull. , :29–63, 2008.[11] A. Sims. Hausdorff ´etale groupoids and their C*-algebras. Operator Algebras andDynamics: Groupoids, Crossed Products, and Rokhlin Dimension , 2020.[12] B. Steinberg. A groupoid approach to discrete inverse semigroup algebras.