Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras
aa r X i v : . [ m a t h . OA ] S e p SELF-SIMILAR GRAPHSA UNIFIED TREATMENT OF KATSURA ANDNEKRASHEVYCH C*-ALGEBRAS
Ruy Exel and Enrique Pardo
Given a graph E , an action of a group G on E , and a G -valued cocycle ϕ on the edgesof E , we define a C*-algebra denoted O G,E , which is shown to be isomorphic to thetight C*-algebra associated to a certain inverse semigroup S G,E built naturally from thetriple (
G, E, ϕ ). As a tight C*-algebra, O G,E is also isomorphic to the full C*-algebra ofa naturally occurring groupoid G tight ( S G,E ). We then study the relationship betweenproperties of the action, of the groupoid and of the C*-algebra, with an emphasis onsituations in which O G,E is a Kirchberg algebra. Our main applications are to Katsuraalgebras and to certain algebras constructed by Nekrashevych from self-similar groups.These two classes of C*-algebras are shown to be special cases of our O G,E , and manyof their known properties are shown to follow from our general theory.
1. Introduction.
The purpose of this paper is to give a unified treatment to two classes of C*-algebraswhich have been studied in the past few years from rather different points of view, namelyKatsura algebras [ ], and certain algebras constructed by Nekrashevych [ ], [ ] fromself-similar groups.The realization that these classes are indeed closely related, as well as the fact thatthey could be given a unified treatment, came to our mind as a result of our attempt tounderstand Katsura’s algebras O A,B from the point of view of inverse semigroups. Thefact, proven by Katsura in [ ], that all Kirchberg algebras in the UCT class may bedescribed in terms of his O A,B was, in turn, a strong motivation for that endeavor.While studying O A,B , it slowly became clear to us that the two matricial parameters A and B , present in Katsura’s construction, play very different roles. The reader acquaintedwith Katsura’s work will easily recognize that the matrix A is destined to be viewed as theedge matrix of a graph, but it took us much longer to realize that B should be thoughtof as providing parameters for an action of the group Z on the graph given by A . Intrying to understand these different roles, some interesting arithmetic popped up sparkinga connection with the work done by Nekrashevych [ ] on the C*-algebra O ( G,X ) associatedto a self-similar group ( G, X ). Date: 27 September 2014.
Mathematics Subject Classification:
Key words and phrases:
Kirchberg algebra, Katsura algebra, Nekrashevych algebra, tight represen-tation, inverse semigroup, groupoid, groupoid C*-algebra.The first-named author was partially supported by CNPq. The second-named author was partiallysupported by PAI III grants FQM-298 and P11-FQM-7156 of the Junta de Andaluc´ıa and by the DGI-MICINN and European Regional Development Fund, jointly, through Project MTM2011-28992-C02-02. r. exel and e. pardo
While Nekrashevych’s algebras contain a Cuntz algebra, Katsura’s algebras contain agraph C*-algebra. This fact alone ought to be considered as a hint that self-similar groupslie in a much bigger class, where the group action takes place on the path space of a graph,rather than on a rooted tree (which, incidentally, is the path space of a bouquet of circles).One of the first important applications of the idea of self-similarity in group theoryis in constructing groups with exotic properties [ ], [ ]. Many of these are defined assubgroups of the group of all automorphisms of a tree. Having been born from automor-phisms, it is natural that the theory of self-similar groups generally assumes that the groupacts faithfully on its tree (see, e.g. [ : Definition 2.1]).However, based on the example provided by Katsura’s algebras, we decided thatperhaps it is best to view the group on its own, the action being an extra ingredient.The main idea behind self-similar groups, namely the equation g ( xw ) = yh ( w ) (1 . : Definition 2.1], and the subsequent notion of restriction , namely g | x := h, depend on faithfulness, since otherwise the group element h appearing in (1.1) would notbe unique and therefore will not be well defined as a function of g and x . Working withnon-faithful group actions we were forced to postulate a functional dependence h = ϕ ( g, x ) , and we were surprised to find that the natural properties expected of ϕ are that of a groupcocycle.To be precise, the ingredients needed in our generalization of self-similar groups are:a countable discrete group G , an action G × E → E of G on a finite graph E = ( E , E , r, d ), and a one-cocycle ϕ : G × E → G for the action of G on the edges of E .Starting with this data (which we assume satisfies a few other natural axioms) weconstruct an action of G on the space of finite paths E ∗ which satisfies the “self-similarity”equation g ( αβ ) = ( gα ) (cid:0) ϕ ( g, α ) β (cid:1) , ∀ g ∈ E, ∀ α, β ∈ E ∗ . Adopting a philosophy similar to that embraced by Katsura and Nekrashevych, wedefine a C*-algebra, denoted O G,E , nverse semigroup actions and self-similar graphs 3 in terms of generators and relations inspired by the above group action. The study of O G,E is, thus, the purpose of this paper.Given a self-similar group (
G, X ), if we consider X as the set of edges of a graph with asingle vertex, and if we define ϕ ( g, x ) = g | x , then our O G,E coincides with Nekrashevych’s O ( G,X ) .On the other hand, if we are given two integer N × N matrices A and B , with A i,j ≥ i and j , we may form a graph E with vertex set E = { , , . . . , N } and with A i,j edges from vertex i to vertex j .We may then use B to define an action of Z on E , by fixing all vertices and acting onthe set of edges as follows: denote the set of edges in E from i to j by { e i,j,n : 0 ≤ n < A i,j } . Given m ∈ Z , and given an edge e i,j,n , in order to define σ m ( e i,j,n ), we first perform theEuclidean division of mB i,j + n by A i,j , say mB i,j + n = ˆ kA i,j + ˆ n with 0 ≤ ˆ n < A i,j . We then put σ m ( e i,j,n ) := e i,j, ˆ n , so that the group element m permutes the A i,j edges from i to j in the same way thataddition by mB i,j , modulo A i,j , permutes the integers { , , . . . , A i,j − } .The quotient ˆ k in the above Euclidean division also plays an important role, namelyin the definition of the cocycle: ϕ ( m, e i,j,n ) := ˆ k. In possession of the graph, the action of Z , and the cocycle ϕ constructed above, weapply our construction and we find that O G,E is isomorphic to Katsura’s O A,B .So, both Nekrashevych’s and Katsura’s algebras become special cases of our construc-tion. We therefore believe that the project of studying such group actions on path spacesas well as the corresponding algebras is of great importance.Taking the first few steps in this direction we have been able to describe O G,E asthe C*-algebra of an ´etale groupoid G G,E , whose construction is remarkably similar to thegroupoid associated to the relation of “tail equivalence with lag” on the path space, asdescribed by Kumjian, Pask, Raeburn and Renault in [ ].The first similarity is that our groupoid G G,E has the exact same unit space as thecorresponding graph groupoid, namely the infinite path space. The second, and mostsurprising similarity is that G G,E is also described by a lag function, except that the valuesof the lag are not integer numbers, as in [ ], but lie in a slightly more complicated group,namely the semi-direct product of the corona group of G by the right shift automorphism(see below for precise definitions).We would like to stress that, like Nekrashevych’s groupoid [ : Theorem 5.1], ourgroupoid G G,E is constructed as a groupoid of germs. However, departing from Nekra-shevych’s techniques, we use Patterson’s [ ] notion of “germs”, rather than the one r. exel and e. pardo employed in [ : Section 5]. While agreeing in many cases, such as when the action istopologically free (see below for the precise definition), the former has a much betterchance of producing Hausdorff groupoids and, in our case, we are able to give a precisecharacterization of Hausdorffness in terms of a property we call pseudo freeness (see belowfor the precise definition).The techniques we use to give O G,E a groupoid model bear heavily on the theory oftight representations of inverse semigroups developed by the first named author in [ ]. Inparticular, from our initial data we construct an abstract inverse semigroup S G,E and showthat O G,E is the universal C*-algebra for tight representations of S G,E .In another direction we again take inspiration from Nekrashevych [ ] and give adescription of O G,E as a Cuntz-Pimsner algebra for a very natural correspondence M overthe algebra C ( E ) ⋊ G. As a result we are able to prove that O G,E is nuclear when G is amenable.We briefly study the natural representation of the graph C*-algebra C ∗ ( E ) [ ] into O G,E , which turns out to be faithful. Also, we study the natural representation of thegroup G into O G,E , which turns out to be faithful when the triple (
G, E, ϕ ) satisfiespseudo freeness, but fails in general.Simplicity of O G,E is also discussed by using our description of this algebra as agroupoid C*-algebra and employing results from [ ]. In doing so, it is crucial to determinewhen is G tight ( S G,E ) a Hausdorff, minimal essentially principal groupoid. To this end, westrongly rely on results obtained by both authors in [ ] about characterization of minimal-ity and essential irreducibility for the groupoid of germs of a general ∗ -inverse semigroup.We then specialize these results to the particular context of the inverse semigroup S G,E .Hence, we characterize Hausdorffness of G tight ( S G,E ) in terms of the existence of finitelymany minimal strongly fixed paths (see below for a precise definition).Also, we characterize minimality of G tight ( S G,E ) in terms of weak G -transitivity ofthe graph (see Section 13 for a definition of this concept). We then obtain a naturalgeneralization of the analog result obtained in [ ] for Exel-Laca algebras.We also show that being essentially principal is related to the topological freeness ofthe action of S G,E on the infinite path space. In this sense, we obtain a characterizationthat relies on the existence of entries for any circuit of the graph, plus a formal conditionwhich forces any element of G fixing open sets to be tighly related to the existence ofsuitable minimal strongly fixed paths.Moreover, we give sufficient conditions on G tight ( S G,E ) to guarantee its local contrac-tiveness (see e.g. [ ] for a definition); this property turns out to be a consequence ofessential principality, so that any simple algebra in the class O G,E will be purely infinitesimple.With the machinery developped we are then able to give a characterization of sim-plicity (and so pure infinite simplicity) for O G,E when G tight ( S G,E ) is Hausdorff.Finally, we revisit the case of Nekrashevych and Katsura algebras, giving a picture ofthe properties enjoyed by these algebras that turns out to be more general than the onesgiven by Nekrashevych or Katsura. nverse semigroup actions and self-similar graphs 5
Some of the results in the present paper appeared in the preprints [ ] and [ ], whichin turn are to be replaced by the present work.We would also like to mention [ ] and [ ], which are strongly related to the algebraswe study here. In [ ] conditions are given for O G,E to be a partial crossed product andin [ ] an interesting connection with Zappa-Sz´ep products is made.Part of this work was done during visits of the second named author to the Departa-mento de Matem´atica da Universidade Federal de Santa Catarina (Florian´opolis, Brasil)and he would like to express his thanks to the host center for its warm hospitality. Bothauthors thank Benjamin Steinberg for interesting discussions on topological freeness ofactions, and Hausdorffness of groupoids.
2. Groups acting on graphs.
Let E = ( E , E , r, d ) be a directed graph, where E denotes the set of vertices , E is the set of edges , r is the range map, and d is the source , or domain map.By definition, a source in E is a vertex x ∈ E , for which r − ( x ) = ∅ . Thus, whenwe say that a graph has no sources , we mean that r − ( x ) = ∅ , for all x ∈ E .By an automorphism of E we shall mean a bijective map σ : E ˙ ∪ E → E ˙ ∪ E such that σ ( E i ) ⊆ E i , for i = 0 ,
1, and moreover such that r ◦ σ = σ ◦ r , and d ◦ σ = σ ◦ d ,on E . It is evident that the collection of all automorphisms of E forms a group undercomposition.By an action of a group G on a graph E we shall mean a group homomorphism from G to the group of all automorphisms of E .If X is any set, and if σ is an action of a group G on X , we shall say that a map ϕ : G × X → G is a one-cocycle for σ , when ϕ ( gh, x ) = ϕ (cid:0) g, σ h ( x ) (cid:1) ϕ ( h, x ) , (2 . g, h ∈ G , and all x ∈ X . In this case, plugging g = h = 1, above, we see thatnecessarily ϕ (1 , x ) = 1 , (2 . x . ◮ Standing Hypothesis.
Throughout this work we shall let G be a countablediscrete group, E be a finite graph with no sources, σ be an action of G on E , and ϕ : G × E → G be a one-cocycle for the restriction of σ to E , which moreover satisfies σ ϕ ( g,e ) ( x ) = σ g ( x ) , ∀ g ∈ G, ∀ e ∈ E , ∀ x ∈ E . (2 . . r. exel and e. pardo The assumptions that E is finite and has no sources will in fact only be used in thenext section and it could probably be removed by using well known graph C*-algebratechniques.By a path in E of length n ≥ α = α α . . . α n , where α i ∈ E , and d ( α i ) = r ( α i +1 ), for all i (this is the usual convention when treatinggraphs from a categorical point of view, in which functions compose from right to left).The range of α is defined by r ( α ) = r ( α ) , while the source of α is defined by d ( α ) = d ( α n ) . A vertex x ∈ E will be considered a path of length zero, in which case we set r ( x ) = d ( x ) = x .For every integer n ≥ E n the set of all paths in E of length n (thisbeing consistent with the already introduced notations for E and E ). Finally, we denoteby E ∗ the sets of all finite paths, and by E ≤ n the set of all paths of length at most n ,namely E ∗ = S k ≥ E k , and E ≤ n = n S k =0 E k . We will often employ the operation of concatenation of paths. That is, if (and onlyif) α and β are paths such that d ( α ) = r ( β ), we will denote by αβ the path obtained byjuxtaposing α and β .In the special case in which α is a path of length zero, the concatenation αβ is allowedif and only if α = r ( β ), in which case we set αβ = β . Similarly, when | β | = 0, then αβ isdefined iff d ( α ) = β , and then αβ = α .We would now like to describe a certain extension of σ and ϕ to finite paths. Under the assumptions of (2.3) there exists a unique pair ( σ ∗ , ϕ ∗ ) ,formed by an action σ ∗ of G on E ∗ (viewed simply as a set), and a one-cocycle ϕ ∗ for σ ∗ ,such that, for every n ≥ , every g ∈ G , and every x ∈ E , one has that: (i) σ ∗ g = σ g , on E ≤ , (ii) ϕ ∗ ( g, x ) = g , (iii) ϕ ∗ = ϕ , on G × E , (iv) σ ∗ g ( E n ) ⊆ E n , (v) r ◦ σ ∗ g = σ g ◦ r , on E n , (vi) d ◦ σ ∗ g = σ g ◦ d , on E n , (vii) σ ϕ ∗ ( g,α ) ( x ) = σ g ( x ) , for all α ∈ E n , nverse semigroup actions and self-similar graphs 7 (viii) σ ∗ is the identity on E n , (ix) σ ∗ g ( αβ ) = σ ∗ g ( α ) σ ∗ ϕ ∗ ( g,α ) ( β ) , provided α and β are finite paths with αβ ∈ E n , (x) ϕ ∗ ( g, αβ ) = ϕ ∗ (cid:0) ϕ ∗ ( g, α ) , β (cid:1) , provided α and β are finite paths with αβ ∈ E n . Proof.
Initially notice that, once (v), (vi) and (vii) are proved, the concatenation of thepaths “ σ ∗ g ( α )” and “ σ ∗ ϕ ∗ ( g,α ) ( β )”, appearing in (ix), is permitted because r (cid:0) σ ∗ ϕ ∗ ( g,α ) ( β ) (cid:1) (v) = σ ϕ ∗ ( g,α ) ( r ( β )) (vii) = σ g ( r ( β )) = σ g (cid:0) d ( α ) (cid:1) (vi) = d (cid:0) σ ∗ g ( α ) (cid:1) . For every g in G , define σ ∗ g on E ≤ to coincide with σ g . Also, define ϕ ∗ on G × E ≤ by (ii) and (iii). It is then clear that (i–iii) hold and it is easy to see that the remainingproperties (iv–x) hold for all n ≤ σ ∗ and ϕ ∗ by induction, so we assume that m ≥ σ ∗ g : E ≤ m → E ≤ m is defined for all g in G , that ϕ ∗ : G × E ≤ m → G, is defined, and that (i–x) hold for all n ≤ m . We then define σ ∗ g : E m +1 → E m +1 for all g in G , and ϕ ∗ : G × E m +1 → G, by induction as follows. Given α ∈ E m +1 , write α = α ′ α ′′ , with α ′ ∈ E , and α ′′ ∈ E m ,and put σ ∗ g ( α ) = σ g ( α ′ ) σ ∗ ϕ ( g,α ′ ) ( α ′′ ), and ϕ ∗ ( g, α ) = ϕ ∗ (cid:0) ϕ ( g, α ′ ) , α ′′ (cid:1) . (2 . . σ g ( α ′ )” and “ σ ∗ ϕ ( g,α ′ ) ( α ′′ )”, appearing above, is permitted. We next verify (iv–x),substituting m + 1 for n .We have that the length of σ ∗ g ( α ), as defined above, is clearly 1 + m , thus proving (iv).With respect to (v) we have that r (cid:0) σ ∗ g ( α ) (cid:1) = r (cid:0) σ g ( α ′ ) (cid:1) = σ g (cid:0) r ( α ′ ) (cid:1) = σ g (cid:0) r ( α ) (cid:1) . As for (vi), notice that d (cid:0) σ ∗ g ( α ) (cid:1) = d (cid:0) σ ∗ ϕ ( g,α ′ ) ( α ′′ ) (cid:1) = σ ϕ ( g,α ′ ) (cid:0) d ( α ′′ ) (cid:1) = σ g (cid:0) d ( α ′′ ) (cid:1) = σ g (cid:0) d ( α ) (cid:1) . This is evidently already included in the statement that σ ∗ is an action, but we repeat it here to aidour proof by induction. r. exel and e. pardo Given x ∈ E , we have that σ ϕ ∗ ( g,α ) ( x ) = σ ϕ ∗ ( ϕ ( g,α ′ ) ,α ′′ ) ( x ) = σ ϕ ( g,α ′ ) ( x ) = σ g ( x ) , taking care of (vii).The verification of (viii) is done as follows: for α = α ′ α ′′ , as in (2.4.1), one has σ ∗ ( α ) = σ ∗ ( α ′ α ′′ ) = σ ( α ′ ) σ ∗ ϕ (1 ,α ′ ) ( α ′′ ) (2.2) = σ ( α ′ ) σ ∗ ( α ′′ ) = α ′ α ′′ = α. In order to prove (ix), pick paths α in E k and β in E l , where k + l = m + 1, and suchthat d ( α ) = r ( β ).We leave it for the reader to verify (ix) in the easy case in which k = 0, that is, when α is a vertex. The case k = 1 is also easy as it is nothing but the definition of σ ∗ g given in(2.4.1). So we may assume that k ≥ α = α ′ α ′′ , with α ′ ∈ E , and α ′′ ∈ E k − , we then have that αβ = α ′ α ′′ β ,and hence, by definition, σ ∗ g ( αβ ) = σ g ( α ′ ) σ ∗ ϕ ( g,α ′ ) ( α ′′ β ) = σ g ( α ′ ) σ ∗ ϕ ( g,α ′ ) ( α ′′ ) σ ∗ ϕ ∗ ( ϕ ( g,α ′ ) ,α ′′ ) ( β ) == σ ∗ g ( α ′ α ′′ ) σ ∗ ϕ ∗ ( g,α ′ α ′′ ) ( β ) . We remark that, in last step above, one should use the induction hypothesis in case k ≤ m ,and the definitions of σ ∗ and ϕ ∗ , when k = m + 1.To verify (x) we again pick paths α in E k and β in E l , where k + l = m + 1, and suchthat d ( α ) = r ( β ). We once more leave the easy case k = 0 to the reader and observe thatthe case k = 1 follows from the definition of ϕ ∗ .We may then suppose that k ≥
2, so we write α = α ′ α ′′ , with α ′ ∈ E , and α ′′ ∈ E k − .Then ϕ ∗ ( g, αβ ) = ϕ ∗ ( g, α ′ α ′′ β ) = ϕ ∗ (cid:0) ϕ ( g, α ′ ) , α ′′ β (cid:1) = ϕ ∗ (cid:16) ϕ ∗ (cid:0) ϕ ( g, α ′ ) , α ′′ (cid:1) , β (cid:17) == ϕ ∗ (cid:16) ϕ ∗ ( g, α ′ α ′′ (cid:1) , β (cid:17) = ϕ ∗ (cid:16) ϕ ∗ ( g, α (cid:1) , β (cid:17) . Let us now prove that σ ∗ is in fact an action of G on E n . We begin by proving that σ ∗ g σ ∗ h = σ ∗ gh on E n , for every g and h in G , which we do by induction on n .This follows immediately from the hypothesis for n ≤
1, so let us assume that n ≥ α ∈ E n , write α = α ′ α ′′ , with α ′ ∈ E , and α ′′ ∈ E n − . Then σ ∗ g (cid:0) σ ∗ h ( α ) (cid:1) = σ ∗ g (cid:0) σ ∗ h ( α ′ α ′′ ) (cid:1) = σ ∗ g (cid:0) σ h ( α ′ ) σ ϕ ( h,α ′ ) ( α ′′ ) (cid:1) == σ g (cid:0) σ h ( α ′ ) (cid:1) σ ∗ ϕ ( g,σ h ( α ′ )) (cid:0) σ ϕ ( h,α ′ ) ( α ′′ ) (cid:1) = σ gh ( α ′ ) σ ∗ ϕ ( g,σ h ( α ′ )) ϕ ( h,α ′ ) ( α ′′ ) == σ gh ( α ′ ) σ ∗ ϕ ( gh,α ′ ) ( α ′′ ) = σ ∗ gh ( α ′ α ′′ ) = σ ∗ gh ( α ) . nverse semigroup actions and self-similar graphs 9 That α ∗ g is bijective on each E n then follows from (viii), so α ∗ is indeed an action of G on E n .Finally, let us show that ϕ ∗ is a cocycle for σ ∗ on E n . For this fix g and h in G andlet α ∈ E n . Then, with α = α ′ α ′′ , as before, ϕ ∗ ( gh, α ) = ϕ ∗ ( gh, α ′ α ′′ ) = ϕ ∗ ( ϕ ( gh, α ′ ) , α ′′ ) = ϕ ∗ (cid:16) ϕ (cid:0) g, σ h ( α ′ ) (cid:1) ϕ ( h, α ′ ) , α ′′ (cid:17) == ϕ ∗ (cid:16) ϕ (cid:0) g, σ h ( α ′ ) (cid:1) , σ ∗ ϕ ( h,α ′ ) ( α ′′ ) (cid:17) ϕ ∗ (cid:0) ϕ ( h, α ′ ) , α ′′ (cid:1) =: ( ⋆ ) . On the other hand, focusing on the right-hand-side of (2.1), notice that ϕ ∗ ( g, σ ∗ h ( α )) ϕ ∗ ( h, α ) = ϕ ∗ (cid:0) g, σ ∗ h ( α ′ α ′′ ) (cid:1) ϕ ∗ ( h, α ′ α ′′ ) == ϕ ∗ (cid:0) g, σ h ( α ′ ) σ ∗ ϕ ( h,α ′ ) ( α ′′ ) (cid:1) ϕ ∗ (cid:0) ϕ ( h, α ′ ) , α ′′ (cid:1) == ϕ ∗ (cid:16) ϕ (cid:0) g, σ h ( α ′ ) (cid:1) , σ ∗ ϕ ( h,α ′ ) ( α ′′ ) (cid:17) ϕ ∗ (cid:0) ϕ ( h, α ′ ) , α ′′ (cid:1) , which coincides with ( ⋆ ) above. This concludes the proof. (cid:3) The only action of G on E ∗ to be considered in this paper is σ ∗ so, from now on, wewill adopt the shorthand notation gα = σ ∗ g ( α ) . Moreover, since ϕ ∗ extends ϕ , we will drop the star decoration and denote ϕ ∗ simply as ϕ . The group law, the cocycle condition, and properties (ii), (v), (vi), (vii), (ix) and (x)of Proposition (2.4) may then be rewritten as follows: For every g and h in G , for every x ∈ E , and for every α and β in E ∗ such that d ( α ) = r ( β ) , one has that (a) ( gh ) α = g ( hα ) , (b) ϕ ( gh, α ) = ϕ (cid:0) g, hα (cid:1) ϕ ( h, α ) , (ii) ϕ ( g, x ) = g , (v) r ( gα ) = gr ( α ) , (vi) d ( gα ) = gd ( α ) , (vii) ϕ ( g, α ) x = gx , (ix) g ( αβ ) = ( gα ) ϕ ( g, α ) β , (x) ϕ ( g, αβ ) = ϕ (cid:0) ϕ ( g, α ) , β (cid:1) . This is why it is useful to include (viii) as a separate statement, since we may now use it to provebijectivity.
It might be worth noticing that if ϕ ( g, α ) = 1, then (2.5.ix) reads “ g ( αβ ) = ( gα ) β ”,which may be viewed as an associativity property. However associativity does not hold ingeneral as ϕ is not always trivial, and hence parentheses must be used.On the other hand parentheses are unnecessary in expressions of the form αgβ , when α, β ∈ E ∗ , and g ∈ G , since the only possible interpretation for this expression is theconcatenation of α with gβ .Another useful property of ϕ is in order. For every g ∈ G , and every α ∈ E ∗ , one has that ϕ ( g − , α ) = ϕ ( g, g − α ) − . Proof.
We have 1 = ϕ (1 , α ) = ϕ ( gg − , α ) = ϕ ( g, g − α ) ϕ ( g − , α ) , from where the conclusion follows. (cid:3)
3. The universal C*-algebra O G,E . As in the above section we fix a graph E , an action of a group G on E , and a one-cocycle ϕ satisfying (2.3).It is our next goal to build a C*-algebra from this data but first let us recall thefollowing notion from [ ]: A Cuntz-Krieger E -family consists of a set { p x : x ∈ E } of mutually orthogonal projections and a set { s e : e ∈ E } of partial isometries, all lying in some C*-algebra, and satisfying(i) s ∗ e s e = p d ( e ) , for every e ∈ E ,(ii) p x = X e ∈ r − ( x ) s e s ∗ e , for every x ∈ E for which r − ( x ) is finite and nonempty. We define O G,E to be the universal unital C*-algebra generated by aset { p x : x ∈ E } ∪ { s e : e ∈ E } ∪ { u g : g ∈ G } , subject to the following relations:(a) { p x : x ∈ E } ∪ { s e : e ∈ E } is a Cuntz-Krieger E -family,(b) the map u : G → O G,E , defined by the rule g u g , is a unitary representation of G ,( c ) u g s e = s ge u ϕ ( g,e ) , for every g ∈ G , and e ∈ E ,(d) u g p x = p gx u g , for every g ∈ G , and x ∈ E . nverse semigroup actions and self-similar graphs 11 Observe that, under our standing assumptions (2.3), for every x ∈ E we have that r − ( x ) is finite and nonempty. So (3.1.ii) and (3.2.a) imply that u g p x u ∗ g = X r ( e )= x u g s e s ∗ e u ∗ g = X r ( e )= x s ge u ϕ ( g,e ) u ∗ ϕ ( g,e ) s ∗ ge == X r ( e )= x s ge s ∗ ge = X r ( f )= gx s f s ∗ f = p gx , which says that (3.2.d) follows from the other conditions. We have nevertheless includedit in (3.2) in the belief that our theory may be generalized to graphs with sources.Our construction generalizes some well known constructions in the literature as wewould now like to mention. Let (
G, X ) be a self similar group as in [ : Definition 2.1]. We may thenconsider a graph E having only one vertex and such that E = X . If we define ϕ ( g, x ) = g | x , where, in the terminology of [ ], g | x is the restriction (or section) of g at x , then the triple( G, E, ϕ ) satisfies (2.3) and one may easily show that O G,E is isomorphic to the algebra O ( G,X ) introduced by Nekrashevych in [ ]. As in [ ], let us assume we are given two N × N matrices A and B withinteger entries, and such that A i,j ≥
0, for all i and j . We may then consider the graph E with vertex set E = { , , . . . , N } , and such that, for each pair of vertices i, j ∈ E , the set of edges from vertex j to vertex i is a set with A i,j elements, say { e i,j,n : 0 ≤ n < A i,j } . Assuming moreover that A has no identically zero rows, it is easy to see that E hasno sources.Define an action σ of Z on E , which is trivial on E , and which acts on edges asfollows: given m ∈ Z , and e i,j,n ∈ E , let (ˆ k, ˆ n ) be the unique pair of integers such that mB i,j + n = ˆ kA i,j + ˆ n , and 0 ≤ ˆ n < A i,j . That is, ˆ k is the quotient and ˆ n is the remainder of the Euclidean division of mB i,j + n by A i,j . We then put σ m ( e i,j,n ) = e i,j, ˆ n . In other words, σ m corresponds to the addition of mB i,j to the variable “ n ” of “ e i,j,n ”,taken modulo A i,j . In turn, the one-cocycle is defined by ϕ ( m, e i,j,n ) = ˆ k. Observe that if A i,j = 0, then there are no edges from j to i , so the value B i,j isentirely irrelevant for the above construction. Therefore it makes no difference to assumethat A i,j = 0 ⇒ B i,j = 0 . It may then be proved without much difficulty that O Z ,E is isomorphic to Katsura’s[ ] algebra O A,B , under an isomorphism sending each u m to the m th power of the unitary u := N X i =1 u i in O A,B , and sending s e i,j,n to s i,j,n .When N = 1, the relevant graph for Katsura’s algebras is the same as the one we usedabove in the description of Nekrashevych’s example. However the former is not a specialcase of the latter because, contrary to what is required in [ ], the group action might notbe faithful. Given any finite graph E , and any action σ of a group G on E , the map ϕ : G × E → G defined by ϕ ( g, a ) = g, ∀ g ∈ G, ∀ a ∈ E is a one-cocycle, and the triple ( G, E, ϕ ) satisfies (2.3). By (3.2.c), we have that u g s a u ∗ g = s ga , for any g in G , and every a in E . It is therefore easy to see that O G,E is isomorphic to thecrossed product of the graph C*-algebra C ∗ ( E ) [ ] by G , relative to the natural actionof G on C ∗ ( E ) induced by σ . In particular, if σ is the trivial action, we have that O G,E isthe maximal tensor product of C ∗ ( E ) by the full group C*-algebra of G . Given any finite graph without sources, and any action σ of a group G on E fixing the vertices, consider the map ϕ : G × E → G defined by ϕ ( g, a ) = 1 , ∀ g ∈ G, ∀ a ∈ E . It is easy to see that ϕ is a one-cocycle, and that the triple ( G, E, ϕ ) satisfies (2.3). Since E has no sources we have, for any g in G , that u g = X x ∈ E u g p x = X x ∈ E X a ∈ r − ( x ) u g s a s ∗ a (3.2.c) = X x ∈ E X a ∈ r − ( x ) s ga s ∗ a , which therefore lies in the copy of C ∗ ( E ) within O G,E . Since the natural representationof C ∗ ( E ) in O G,E is faithful by (11.1), the conclusion is that O G,E ∼ = C ∗ ( E ). nverse semigroup actions and self-similar graphs 13 We now return to the general case of a triple (
G, E, ϕ ) satisfying (2.3). We initiallyrecall the usual extension of the notation “ s e ” to allow for paths of arbitrary length. Given a finite path α in E ∗ , we shall let s α denote the element of O G,E given by: (i) when α = x ∈ E , we let s α = p x , (ii) when α ∈ E , then s α is already defined above, (iii) when α ∈ E n , with n > , write α = α ′ α ′′ , with α ′ ∈ E , and α ′′ ∈ E n − , and set s α = s α ′ s α ′′ , by recurrence. Commutation relation (3.2.c) may then be generalized to finite paths as follows:
Given α ∈ E ∗ , and g ∈ G , one has that u g s α = s gα u ϕ ( g,α ) . Proof.
Let n be the length of α . When n = 0 ,
1, this follows from (3.2.d&c), respectively.When n >
1, write α = α ′ α ′′ , with α ′ ∈ E , and α ′′ ∈ E n − . Using induction, we thenhave u g s α = u g s α ′ s α ′′ = s gα ′ u ϕ ( g,α ′ ) s α ′′ = s gα ′ s ϕ ( g,α ′ ) α ′′ u ϕ ( ϕ ( g,α ′ ) ,α ′′ ) == s ( gα ′ ) ϕ ( g,α ′ ) α ′′ u ϕ ( g,α ′ α ′′ ) = s g ( α ′ α ′′ ) u ϕ ( g,α ′ α ′′ ) = s gα u ϕ ( g,α ) . (cid:3) Our next result provides a spanning set for O G,E . Let S = (cid:8) s α u g s ∗ β : α, β ∈ E ∗ , g ∈ G, d ( α ) = gd ( β ) (cid:9) ∪ { } . Then S is closed under multiplication and adjoints and its closed linear span coincideswith O G,E . Proof.
That S is closed under adjoints is clear. With respect to closure under multiplica-tion, let s α u g s ∗ β and s γ u h s ∗ δ be elements of S .From (3.2.a) we know that s ∗ β s γ = 0, unless either γ = βε , or β = γε , for some ε ∈ E ∗ .If γ = βε , then s ∗ β s γ = s ∗ β s βε = s ∗ β s β s ε = s ε , and hence( s α u g s ∗ β )( s γ u h s ∗ δ ) = s α u g s ε u h s ∗ δ = s α s gε u ϕ ( g,ε ) u h s ∗ δ = s αgε u ϕ ( g,ε ) h s ∗ δ . (3 . . d ( αgε ) = d ( gε ) = gd ( ε ) = ϕ ( g, ε ) d ( ε ) = ϕ ( g, ε ) d ( γ ) = ϕ ( g, ε ) hd ( δ ) , we deduce that the element appearing in the right-hand-side of (3.9.1) indeed belongs to S . In the second case, namely if β = γε , then the adjoint of the term appearing in theleft-hand-side of (3.9.1) is ( s δ u h − s ∗ γ )( s β u g − s ∗ α ) , and the case already dealt with implies that this belongs to S . The result then followsfrom the fact that S is self-adjoint.In order to prove that O G,E coincides with the closed linear span of S , let A denotethe latter. Given that S is self-adjoint and closed under multiplication, we see that A is aclosed *-subalgebra of O G,E . Since A evidently contains s α for every α in E ≤ , and sinceit also contains u g for every g in G , we deduce that A = O G,E . (cid:3)
4. The inverse semigroup S G,E . As before, we keep (2.3) in force.In this section we will give an abstract description of the set S appearing in (3.9)as well as its multiplication and adjoint operation. The goal is to construct an inversesemigroup from which we will later recover O G,E . Over the set S G,E = (cid:8) ( α, g, β ) ∈ E ∗ × G × E ∗ : d ( α ) = gd ( β ) (cid:9) ∪ { } , consider a binary multiplication operation defined by( α, g, β )( γ, h, δ ) = ( αgε, ϕ ( g, ε ) h, δ ) , if γ = βε, ( α, gϕ ( h − , ε ) − , δh − ε ) , if β = γε, , otherwise,and a unary adjoint operation defined by( α, g, β ) ∗ := ( β, g − , α ) . Furthermore, the subset of S G,E formed by all elements ( α, g, β ), with g = 1, will bedenoted by S E .It is easy to see that S E is closed under the above operations, and that it is isomorphicto the inverse semigroup generated by the canonical partial isometries in the graph C*-algebra of E .Let us begin with a simple, but useful result: Given ( α, g, β ) and ( γ, h, δ ) in S G,E , one has β = γ ⇒ ( α, g, β )( γ, h, δ ) = ( α, gh, δ ) . Proof.
Focusing on the first clause of (4.1), write γ = βε , with ε = d ( β ). Then( α, g, β )( γ, h, δ ) = ( αgd ( β ) , ϕ (cid:0) g, d ( β ) (cid:1) h, δ ) = ( αd ( α ) , gh, δ ) = ( α, gh, δ ) . (cid:3) S G,E is an inverse semigroup with zero. nverse semigroup actions and self-similar graphs 15
Proof.
We leave it for the reader to prove that the above operations are well defined andthe multiplication is associative. In order to prove the statement it then suffices [ :Theorem 1.1.3] to show that, for all y, z ∈ S G,E , one has that(i) yy ∗ y = y , and(ii) yy ∗ commutes with zz ∗ .Given y = ( α, g, β ) ∈ S G,E , we have by the above Lemma that yy ∗ y = ( α, g, β )( β, g − , α )( α, g, β ) = ( α, , α )( α, g, β ) = ( α, g, β ) = y, proving (i). Notice also that yy ∗ = ( α, , α ) (4 . . S E , which is a commutative set because S E isan inverse semigroup. Point (ii) above then follows immediately, concluding the proof. (cid:3) As seen in (4.3.1), the idempotent semi-lattice of S G,E , henceforth denoted by E , isgiven by E = (cid:8) ( α, , α ) : α ∈ E ∗ (cid:9) ∪ { } . (4 . E is also the idempotent semi-lattice of S E .For simplicity, from now on we will adopt the short-hand notation f α = ( α, , α ) , ∀ α ∈ E ∗ . (4 . If α, β ∈ E ∗ , then f α f β = f α , if there exists γ such that α = βγ,f β , if there exists γ such that αγ = β, , otherwise. Recall that if α and β are in E ∗ , we say that α (cid:22) β , if α is a prefix of β , i.e. if thereexists γ ∈ E ∗ , such that αγ = β . It therefore follows from (4.6) that f α ≤ f β ⇐⇒ β (cid:22) α. (4 . e, f ∈ E , one has thateither e ⊥ f , or e and f are comparable. It follows that e ⋓ f ⇒ e ≤ f , or f ≤ e. (4 .
5. Pseudo freeness and E*-unitarity.
Again working under (2.3), suppose we are given g in G and a finite path α such that gα = α , and ϕ ( g, α ) = 1 . (5 . α ′ extending α , that is a path of the form α ′ = αβ , where β is another finite path, we have gα ′ = g ( αβ ) = ( gα ) ϕ ( g, α ) β = αβ = α ′ , and ϕ ( g, α ′ ) = ϕ ( g, αβ ) = ϕ (cid:0) ϕ ( g, α ) , β (cid:1) = ϕ (cid:0) , β (cid:1) = 1 . This says that any path α ′ extending α also satisfies (5.1) so, in particular, every extensionof α is fixed by g . If g ∈ G and α is a finite path satisfying (5.1), we will say that α is strongly fixed by g . In addition, if no proper prefix of α is strongly fixed by g , we will saythat α is a minimal strongly fixed path for g .The following result is an easy consequence of the discussion above: Given g in G , let M g be the set of all minimal strongly fixed pathsfor g . Then the set of all strongly fixed paths for g is given by F µ ∈ M g { µγ : γ ∈ E ∗ , d ( µ ) = r ( γ ) } , where the square cup stands for disjoint union. Let us now introduce terminology to describe situations in which nontrivial stronglyfixed paths do not exist.
We will say that (
G, E, ϕ ) is pseudo free if, whenever ( g, e ) ∈ G × E ,is such that ge = e , and ϕ ( g, e ) = 1, then g = 1.Notice that pseudo freeness is equivalent to the fact that an edge is never a stronglyfixed path for a nontrivial group element. In fact we may boost this up to finite paths asfollows: Suppose that ( G, E, ϕ ) is pseudo free and that a finite path α ofnonzero length is strongly fixed for some g in G . Then g = 1 . In a preprint version of this work we have used the term residually free to refer to the conceptpresently being defined, but this apparently conflicts with a well established notion in group theory. nverse semigroup actions and self-similar graphs 17
Proof.
Arguing by contradiction, assume that there is a counter-example to the statement,meaning that there is a strongly fixed path α for a nontrivial group element g . Then, asalready mentioned, α has a minimal strongly fixed prefix, so we may assume without lossof generality that α itself is minimal.By (2.5.ii), α can’t be a vertex, and neither can it be an edge, by hypothesis. So | α | ≥
2, and we may then write α = βγ , with β, γ ∈ E ∗ , and | β | , | γ | < | α | . Then βγ = α = gα = g ( βγ ) = ( gβ ) ϕ ( g, β ) γ, whence β = gβ , and γ = ϕ ( g, β ) γ , by length considerations. Should ϕ ( g, β ) = 1, the pair( g, β ) would be a smaller counter-example to the statement, violating the minimality of α .So we have that ϕ ( g, β ) = 1. In addition, ϕ (cid:0) ϕ ( g, β ) , γ (cid:1) = ϕ ( g, βγ ) = ϕ ( g, α ) = 1 . It follows that (cid:0) ϕ ( g, β ) , γ (cid:1) is a counter-example to the statement, again violating the min-imality of α . This is a contradiction and hence no counter-example exists whatsoever,concluding the proof. (cid:3) An apparently stronger version of pseudo freeness is in order.
Suppose that ( G, E, ϕ ) is pseudo free. Then, for all g , g ∈ G , and α ∈ E ∗ , one has that g α = g α and ϕ ( g , α ) = ϕ ( g , α ) ⇒ g = g . Proof.
Defining g = g − g , observe that gα = α, and we claim that ϕ ( g, α ) = 1. In fact, ϕ ( g, α ) = ϕ (cid:0) g − g , α (cid:1) = ϕ (cid:0) g − , g α (cid:1) ϕ (cid:0) g , α (cid:1) (2.6) == ϕ (cid:0) g , g − g α (cid:1) − ϕ ( g , α ) = ϕ ( g , α ) − ϕ ( g , α ) = 1 , so it follows that g = 1, which is to say that g = g . (cid:3) We will now determine conditions under which S G,E is E*-unitary. In order to do sowe first need to understand when does an element s of S G,E dominate a nonzero idempotent e , which in turn must necessarily have the form e = ( γ, , γ ), as seen in (4.3.1). If s indeeddominates a nonzero idempotent, it is clear that s is itself nonzero, so s must have theform ( α, g, β ). Let α , β and γ be finite paths in E , and let g ∈ G be such that d ( α ) = gd ( β ) , so that s := ( α, g, β ) is a general nonzero element of S G,E and e := ( γ, , γ ) is a general nonzero idempotent element of S G,E . Then e ≤ s , if and only (i) α = β , (ii) γ = ατ , for some finite path τ , (iii) τ is strongly fixed by g . Proof.
In order to prove the “if” part, we have se = ( α, g, β )( γ, , γ ) = ( α, g, α )( ατ, , γ ) = ( αgτ, ϕ ( g, τ ) , γ ) == ( ατ, , γ ) = ( γ, , γ ) = e, proving that e ≤ s . Conversely, assuming that e ≤ s , we have se = e , so in particular se = 0, and hence by the definition of the multiplication on S G,E , either γ is a prefix of β or vice versa.In case β is a prefix of γ , we may write γ = βτ , for some finite path τ , and then( γ, , γ ) = e = se = ( α, g, β )( βτ, , γ ) = ( αgτ, ϕ ( g, τ ) , γ ) , so we conclude that αgτ = γ = βτ , and ϕ ( g, τ ) = 1 . So α = β , gτ = τ and the statement is proved.On the other hand, if γ is a prefix of β , we may write β = γε and, again according tothe definition of the multiplication on S G,E , the third coordinate of the product se will be γε , from where we conclude that γ = γε . So | ε | = 0 and then γ = β , which in particularmeans that β is a prefix of γ , and the proof follows as above. (cid:3) S G,E is an E*-unitary inverse semigroup if and only if ( G, E, ϕ ) ispseudo free. Proof.
Let s be an element of S G,E which dominates a nonzero idempotent element e . Asdiscussed above, we necessarily have s = ( α, g, β ), and e = ( γ, , γ ) , where α , β and γ are finite paths in E , and d ( α ) = gd ( β ). Then, by (5.7) we concludethat gτ = τ , and ϕ ( g, τ ) = 1 so, assuming that ( G, E, ϕ ) is pseudo free, we have g = 1.Moreover by (5.7.i) we see that α = β , so s = ( α, g, β ) = ( α, , α ) , which is idempotent as desired. In order to prove the converse, let ( g, e ) ∈ G × E , be suchthat ge = e , and ϕ ( g, e ) = 1. Then the element s := (cid:0) d ( e ) , g, d ( e ) (cid:1) lies in S G,E because gd ( e ) = d ( ge ) = d ( e ) . Moreover observe that s dominates the nonzero idempotent element ( e, , e ), since s ( e, , e ) = (cid:0) d ( e ) , g, d ( e ) (cid:1)(cid:0) d ( e ) e, , e (cid:1) = ( d ( e ) ge, ϕ ( g, e ) , e ) = ( e, , e ) . So, under the hypothesis that S G,E is E*-unitary, we conclude that s is idempotent, whichis to say that g = 1. This proves that ( G, E, ϕ ) is pseudo free. (cid:3) nverse semigroup actions and self-similar graphs 19
6. Tight representations of S G,E . As before, we keep (2.3) in force.It is the main goal of this section to show that O G,E is the universal C*-algebra fortight representations of S G,E .Recall from (4.6) that f α ≤ f d ( α ) , for every α ∈ E ∗ , so we see that the set { f x : x ∈ E } (6 . cover [ : Definition 11.5] for E . The map π : S G,E → O
G,E , defined by π (0) = 0 , and π ( α, g, β ) = s α u g s ∗ β , is a tight [ : Definition 13.1] representation. Proof.
We leave it for the reader to show that π is in fact multiplicative and that itpreserves adjoints.In order to prove that π is tight, we shall use the characterization given in [ : Propo-sition 11.8], observing that π satisfies condition (i) of [ : Proposition 11.7] because, withrespect to the cover (6.1), we have that _ x ∈ E π ( f x ) = _ x ∈ E π ( x, , x ) = _ x ∈ E p x = X x ∈ E p x = 1 , by (3.2.a). So we assume that { f α , . . . , f α n } is a cover for a given f β , where α , . . . α n , β ∈ E ∗ , and we need to show that n _ i =1 π ( f α i ) ≥ π ( f β ) . (6 . . i , we have that f α i ≤ f β , so by (4.7) there exists γ i ∈ E ∗ suchthat α i = βγ i .We shall prove (6.2.1) by induction on the variable L = min ≤ i ≤ n | γ i | . If L = 0, we may pick i such that | γ i | = 0, and then necessarily γ i = d ( β ), in whichcase α i = β , and (6.2.1) is trivially true.Assuming that L ≥
1, let x := d ( β ). Observe that x is not a source either becausethis is part of our standing assumptions (2.3), or simply because x is the range of every γ i . In any case let us write r − ( x ) = { e , . . . , e k } , and observe that π ( f β ) = s β s ∗ β = s β p x s ∗ β (3.2.a) = k X j =1 s β s e j s ∗ e j s ∗ β = k X j =1 π ( f βe j ) . In order to prove (6.2.1) it is therefore enough to show that n _ i =1 π ( f α i ) ≥ π ( f βe j ) , (6 . . j = 1 , . . . , k . Fixing j we claim that f βe j is covered by the set Z = (cid:8) f α i : 1 ≤ i ≤ n, f α i ≤ f βe j (cid:9) . In order to see this let y be a nonzero element in E such that y ≤ f βe j . Then y ≤ f β ,and so y ⋓ f α i for some i . Thus, to prove the claim it is enough to check that f α i lies in Z . Observe that yf βe j f α i = yf α i = 0 , which implies that f βe j ⋓ f α i .By (4.8) we have that f βe j and f α i are comparable, so either βe j (cid:22) α i or α i (cid:22) βe j ,by (4.7). Since we are under the hypothesis that L ≥
1, and hence that | α i | = | β i | + | γ i | ≥ | β | + 1 = | βe j | , we must have that βe j (cid:22) α i , from where we deduce that f α i ≤ f βe j , proving our claim.Employing the induction hypothesis we then deduce that _ z ∈ Z π ( z ) ≥ π ( f βe j ) , verifying (6.2.2), and thus concluding the proof. (cid:3) We would now like to prove that the representation π above is in fact the universal tight representation of S G,E . Let A be a unital C*-algebra and let ρ : S G,E → A be a tight represen-tation. Then there exists a unique unital *-homomorphism ψ : O G,E → A , such that thediagram S G,E π −→ O G,E y ψA ................................................................................................................ .......... ρ commutes. nverse semigroup actions and self-similar graphs 21 Proof.
We will initially prove that the elements˜ p x := ρ ( x, , x ) , ∀ x ∈ E , ˜ s e := ρ (cid:0) e, , d ( e ) (cid:1) , ∀ e ∈ E , ˜ u g := X x ∈ E ρ ( x, g, g − x ) , ∀ g ∈ G, satisfy relations (3.2.a–d). Since the f x defined in (4.5) are mutually orthogonal idempo-tents in S G,E , it is clear that the ˜ p x are mutually orthogonal projections. Evidently the ˜ s e are partial isometries so, in order to check (3.2.a), we must only verify (3.1.i) and (3.1.ii).With respect to the former, let e ∈ E . Then˜ s ∗ e ˜ s e = ρ (cid:0) ( d ( e ) , , e )( e, , d ( e ) (cid:1) = ρ (cid:0) d ( e ) , , d ( e ) (cid:1) = ˜ p d ( e ) , proving (3.1.i). In order to prove (3.1.ii), let x be a vertex such that r − ( x ) is nonemptyand write r − ( x ) = (cid:8) e , . . . , e n (cid:9) . Putting q i = ( e i , , e i ), we then claim that the set (cid:8) q , . . . , q n (cid:9) is a cover for q := ( x, , x ). In order to prove this we must show that, if the nonzeroidempotent f is dominated by q , then f ⋓ q i for some i .Let f = ( α, , α ) by (4.4) and notice that0 = f = f q = ( α, , α )( x, , x ) . So α and x are comparable, and this can only happen when x = r ( α ). If | α | = 0 thennecessarily α = x , so f = q , and it is clear that f ⋓ q i for all i . On the other hand, if | α | ≥
1, we write α = α ′ α ′′ , with α ′ ∈ E , so that r ( α ′ ) = r ( α ) = x , and hence α ′ = e i , for some i . Therefore f q i = ( α, , α )( e i , , e i ) = ( α, , α )( α ′ , , α ′ ) = ( α, , α ) = 0 , so f ⋓ q i , proving the claim. Since ρ is a tight representation, we deduce that ρ ( q ) = n _ i =1 ρ ( q i ) , but since the q i are easily seen to be pairwise orthogonal, their supremum coincides withtheir sum, whence ˜ p x = ρ ( q ) = n X i =1 ρ ( q i ) = n X i =1 ρ ( e i , , e i ) = = n X i =1 ρ (cid:0) ( e i , , d ( e i )) ( d ( e i ) , , e i ) (cid:1) = n X i =1 ˜ s e i ˜ s ∗ e i , thus verifying (3.1.ii), and hence proving (3.2.a).With respect to (3.2.b), let us first prove that ˜ u = 1. Considering the subsets of E given by X = ∅ , Y = ∅ , and Z = (cid:8) ( x, , x ) : x ∈ E (cid:9) , notice that, according to [ : Definition 11.4], one has that E X,Y = E , and that Z is a cover for E X,Y , as seen in (6.1). By the tightness condition [ : Definition11.6] we have _ z ∈ Z ρ ( z ) ≥ ^ x ∈ X ρ ( x ) ∧ ^ y ∈ Y ¬ ρ ( y ) . As explained in the discussion following [ : Definition 11.6], the right-hand-side above mustbe interpreted as 1 because X and Y are empty. On the other hand, since the ρ ( z ) arepairwise orthogonal, the supremum in the left-hand-side above becomes a sum, so1 = X z ∈ Z ρ ( z ) = X x ∈ E ρ ( x, , x ) = ˜ u . In order to prove that ˜ u is multiplicative, let g and h be in G . Then˜ u g ˜ u h = X x,y ∈ E ρ (cid:0) ( x, g, g − x )( y, h, h − y ) (cid:1) == X x ∈ E ρ (cid:0) ( x, g, g − x )( g − x, h, h − g − x ) (cid:1) = X x ∈ E ρ (cid:0) x, gh, ( gh ) − x (cid:1) = ˜ u gh . We next claim that ˜ u ∗ g = ˜ u g − , for all g in G . To prove it we compute˜ u ∗ g = X x ∈ E ρ ( x, g, g − x ) ∗ = X x ∈ E ρ ( g − x, g − , x ) = · · · which, upon the change of variables y = g − x , becomes · · · = X y ∈ E ρ ( y, g − , gy ) = ˜ u g − . This shows that ˜ u is a unitary representation, verifying (3.2.b). Turning now ourattention to (3.2.c), let g ∈ G and e ∈ E . Then˜ u g ˜ s e = X x ∈ E ρ ( x, g, g − x ) ρ (cid:0) e, , d ( e ) (cid:1) = ρ (cid:0) gr ( e ) , g, r ( e ) (cid:1) ρ (cid:0) e, , d ( e ) (cid:1) = nverse semigroup actions and self-similar graphs 23 = ρ (cid:0) r ( ge ) ge, ϕ ( g, e ) , d ( e ) (cid:1) = ρ (cid:0) ge, ϕ ( g, e ) , d ( e ) (cid:1) = ( ⋆ ) . On the other hand˜ s ge ˜ u ϕ ( g,e ) = ρ (cid:0) ge, , d ( ge ) (cid:1) X x ∈ E ρ (cid:0) x, ϕ ( g, e ) , ϕ ( g, e ) − x (cid:1) == ρ (cid:0) ge, , d ( ge ) (cid:1) ρ (cid:0) d ( ge ) , ϕ ( g, e ) , g − d ( ge ) (cid:1) == ρ (cid:0) ge, , d ( ge ) (cid:1) ρ (cid:0) d ( ge ) , ϕ ( g, e ) , d ( e ) (cid:1) = ρ (cid:0) ge, ϕ ( g, e ) , d ( e ) (cid:1) , which coincides with ( ⋆ ) and hence proves (3.2.c). We leave the proof of (3.2.d) to thereader after which the universal property of O G,E intervenes to provide us with a *-homomorphism ψ : O G,E → A sending p x ˜ p x , s e ˜ s e , and u g ˜ u g . Now we must show that ψ (cid:0) π ( γ ) (cid:1) = ρ ( γ ) , ∀ γ ∈ S G,E . (6 . . γ = ( x, , x ), for x ∈ E ,(ii) γ = (cid:0) e, , d ( e ) (cid:1) , for e ∈ E ,(iii) γ = ( x, g, g − x ), for x ∈ E , and g ∈ G .In case (i) we have ψ ( π ( γ )) = ψ ( π ( x, , x )) = ψ ( p x ) = ˜ p x = ρ ( x, , x ) = ρ ( γ ) . As for (ii), ψ ( π ( γ )) = ψ (cid:0) π (cid:0) e, , d ( e ) (cid:1)(cid:1) = ψ ( s e ) = ˜ s e = ρ (cid:0) e, , d ( e ) (cid:1) = ρ ( γ ) . Under (iii), ψ ( π ( γ )) = ψ (cid:0) π ( x, g, g − x ) (cid:1) = ψ ( p x u g p g − x ) = ψ ( p x u g ) = ˜ p x ˜ u g == ρ ( x, , x ) X y ∈ E ρ ( y, g, g − y ) = X y ∈ E ρ (cid:0) ( x, , x )( y, g, g − y ) (cid:1) = ρ ( x, g, g − x ) = ρ ( γ ) . In order to prove (6.3.1), it is now clearly enough to check that the *-sub-semigroupof S G,E generated by the elements mentioned in (i–iii), above, coincides with S G,E . Denoting this *-sub-semigroup by T , we will first show that (cid:0) α, , d ( α ) (cid:1) is in T , forevery α ∈ E ∗ . This is evident for | α | ≤
1, so we suppose that α = α ′ α ′′ , with α ′ ∈ E ,and r ( α ′′ ) = d ( α ′ ). We then have by induction that T ∋ (cid:0) α ′ , , d ( α ′ ) (cid:1)(cid:0) α ′′ , , d ( α ′′ ) (cid:1) = (cid:0) α ′ α ′′ , , d ( α ′′ ) (cid:1) = (cid:0) α, , d ( α ) (cid:1) . Considering a general element ( α, g, β ) ∈ S G,E , let x = d ( α ), so that g − x = d ( β ),and notice that T ∋ (cid:0) α, , d ( α ) (cid:1) ( x, g, g − x ) (cid:0) β, , d ( β ) (cid:1) ∗ == (cid:0) α, , d ( α ) (cid:1)(cid:0) d ( α ) , g, d ( β ) (cid:1)(cid:0) d ( β ) , , β (cid:1) = ( α, g, β ) , which proves that T = S G,E , and hence that (6.3.1) holds.To conclude we observe that the uniqueness of ψ follows from the fact that O G,E isgenerated by the p x , the s e , and the u g . (cid:3) Given an inverse semigroup S with zero, recall from [ : Theorem 13.3] that G tight ( S )(denoted simply as G tight in [ ]) is the groupoid of germs for the natural action of S on thespace of tight filters over its idempotent semi-lattice. Moreover the C*-algebra of G tight ( S )is universal for tight representations of S . Under the assumptions of (2.3) one has that O G,E is isomorphic to theC*-algebra of the groupoid G tight ( S G,E ) . Proof.
Follows from [ : Theorem 13.3] and the uniqueness of universal C*-algebras. (cid:3) We should notice that our requirement that G be countable in (2.3) is only used inthe above proof, since the application of [ : Theorem 13.3] depends on the countability of S G,E .
7. The Lag Group.
It is our next goal to give a concrete description of G tight ( S G,E ), similar to the descriptiongiven of the groupoid associated to a row-finite graph in [ : Definition 2.3]. The crucialingredient there is the notion of tail equivalence with lag . In this section we will constructa group where our generalized lag function will take values.Let G be a group. Within the infinite cartesian product G ∞ = Y n ∈ N G consider the infinite direct sum G ( ∞ ) = M n ∈ N G formed by the elements g = ( g n ) n ∈ N ∈ G ∞ which are eventually trivial, that is, for whichthere exists n such that g n = 1, for all n ≥ n . It is clear that G ( ∞ ) is a normal subgroupof G ∞ . For the purpose of this cartesian product we adopt the convention that N = { , , , . . . } . nverse semigroup actions and self-similar graphs 25 Given a group G , the corona of G is the quotient group˘ G = G ∞ /G ( ∞ ) . Consider the left and right shift endomorphisms of G ∞ λ, ρ : G ∞ → G ∞ given for every g = ( g n ) n ∈ N ∈ G ∞ , by λ ( g ) n = g n +1 , ∀ n ∈ N , and ρ ( g ) n = (cid:26) , if n = 0 , g n − , if n ≥ . It is readily seen that G ( ∞ ) is invariant under both λ and ρ , so these pass to the quotientproviding endomorphisms ˘ λ, ˘ ρ : ˘ G → ˘ G. (7 . g = ( g n ) n ∈ N ∈ G ∞ , we have that λ ( ρ ( g )) = g , and ρ ( λ ( g )) = (1 , g , g , . . . ) ≡ g , (7 . ≡ ” to refer to the equivalence relation determined by the normal subgroup G ( ∞ ) . Therefore both ˘ λ ˘ ρ and ˘ ρ ˘ λ coincide with the identity, and hence ˘ λ and ˘ ρ are eachother’s inverse. In particular, they are both automorphisms of ˘ G .Iterating ˘ ρ therefore gives an action of Z on ˘ G . Given any countable discrete group G , the lag group associated to G isthe semi-direct product group ˘ G ⋊ ˘ ρ Z . The reason we call this the “lag group” is that it will play a very important role inthe next section, as the co-domain for our lag function.
8. The tight groupoid of S G,E . We would now like to give a detailed description of the groupoid G tight ( S G,E ). As alreadymentioned this is the groupoid of germs for the natural action of S G,E on the space of tightfilters over the idempotent semi-lattice E of S G,E . See [ : Section 4] for more details.By an infinite path in E we shall mean any infinite sequence of the form ξ = ξ ξ . . . , where ξ i ∈ E , and d ( ξ i ) = r ( ξ i +1 ), for all i . The set of all infinite paths will be denotedby E ∞ . Given an infinite path ξ = ξ ξ . . . ∈ E ∞ , and an integer n ≥
0, denote by ξ | n the finite path of length n given by ξ | n = ξ ξ . . . ξ n , if n ≥ ,r ( ξ ) , if n = 0 . There is a unique action ( g, ξ ) ∈ G × E ∞ gξ ∈ E ∞ of G on E ∞ such that, ( gξ ) | n = g ( ξ | n ) , for every g ∈ G , ξ ∈ E ∞ , and n ∈ N . Proof.
Left to the reader. (cid:3)
Recall from (4.5) that, for any finite path α ∈ E ∗ , we denote by f α the idempotentelement ( α, , α ) in E . Thus, given an infinite path ξ ∈ E ∞ , we may look at the subset F ξ = { f ξ | n : n ∈ N } ⊆ E , which turns out to be an ultra-filter [ : Definition] over E . Denoting the set of all ultra-filters over E by b E ∞ , as in [ : Definition 12.8], one may also show [ : Proposition 19.11]that the correspondence ξ ∈ E ∞
7→ F ξ ∈ b E ∞ is bijective, and we will use it to identify E ∞ and b E ∞ . Furthermore, this correspondencemay be proven to be a homeomorphism if E ∞ is equipped with the product topology.Since E is finite, E ∞ is compact by Tychonov’s Theorem, and consequently so is b E ∞ .Being the closure of b E ∞ within b E [ : Theorem 12.9], the space b E tight formed by the tightfilters therefore necessarily coincides with b E ∞ .Identifying b E tight with E ∞ , as above, we may transfer the canonical action of S G,E from the former to the latter resulting in the following: to each element ( α, g, β ) ∈ S G,E ,we associate the partial homeomorphism of E ∞ whose domain is the cylinder Z ( β ) := { η ∈ E ∞ : η = βξ, for some ξ ∈ E ∞ } , (8 . η = βξ ∈ Z ( β ) to αgξ , where the meaning of “ gξ ” is as in (8.1).As before we will not use any special symbol to indicate this action, using modulenotation instead:( α, g, β ) η = αgξ, ∀ ( α, g, β ) ∈ S G,E , ∀ η = βξ ∈ Z ( β ) . (8 . αgξ is in fact an element of E ∞ , whichis to say that d ( α ) = r ( gξ ) . Firstly, for every element ( α, g, β ) ∈ S G,E , we have that d ( α ) = gd ( β ). Secondly, if η = βξ ∈ E ∞ , then d ( β ) = r ( ξ ). Therefore r ( gξ ) = gr ( ξ ) = gd ( β ) = d ( α ) . This leads to a first, more or less concrete description of G tight ( S G,E ). nverse semigroup actions and self-similar graphs 27 Under (2.3), one has that G tight ( S G,E ) is isomorphic to the groupoidof germs for the above action of S G,E on E ∞ . Our aim is nevertheless a much more precise description of this groupoid. Recall from[ : Definition 4.6] that the germ of an element s ∈ S G,E at a point ξ in the domain of s is denoted by [ s, ξ ]. If s = ( α, g, β ), this would lead to the somewhat awkward notation[( α, g, β ) , ξ ], which from now on will instead be written as (cid:2) α, g, β ; ξ (cid:3) . Thus the groupoid G tight ( S G,E ), consisting of all germs for the action of S G,E on E ∞ ,is given by G tight ( S G,E ) = n(cid:2) α, g, β ; ξ (cid:3) : ( α, g, β ) ∈ S G,E , ξ ∈ Z ( β ) o . (8 . Suppose that ( G, E, ϕ ) is pseudo free and let us be given elements ( α , g , β ) and ( α , g , β ) in S G,E , with | β | ≤ | β | , as well as infinite paths η in Z ( β ) ,and η in Z ( β ) . Then (cid:2) α , g , β ; η (cid:3) = (cid:2) α , g , β ; η (cid:3) if and only if there is a finite path γ and an infinite path ξ , such that (i) α = α g γ, (ii) g = ϕ ( g , γ ) , (iii) β = β γ, (iv) η = η = β γξ. Proof.
Assuming that the germs are equal, we have by [ : Definition 4.6] that η = η =: η, and there is an idempotent ( δ, , δ ) ∈ E , such that η ∈ Z ( δ ), and( α , g , β )( δ, , δ ) = ( α , g , β )( δ, , δ ) . (8 . . η = δζ , for some ζ ∈ E ∞ . Upon replacing δ by a longer prefix of η , wemay assume that | δ | is as large as we want. Furthermore the element of S G,E represented bythe two sides of (8.6.1) is evidently nonzero because the partial homeomorphism associatedto it under our action has η in its domain. So, focusing on (4.1), we see that β and δ arecomparable, and so are β and δ .Assuming that | δ | exceeds both | β | and | β | , we may then write δ = β ε = β ε ,for suitable ε and ε in E ∗ . But since | β | ≤ | β | , this in turn implies that β = β γ , forsome γ ∈ E ∗ , hence proving (iii). Therefore δ = β γε , so η = δζ = β γε ζ, and (iv) follows once we choose ξ = ε ζ . Moreover, equation (8.6.1) reads( α , g , β )( β γε , , β γε ) = ( α , g , β γ )( β γε , , β γε ) . Computing the products according to (4.1), we get (cid:0) α g ( γε ) , ϕ ( g , γε ) , β γε (cid:1) = (cid:0) α g ε , ϕ ( g , ε ) , β γε (cid:1) , from where we obtain α g ε = α g ( γε ) = α ( g γ ) ϕ ( g , γ ) ε , (8 . . ϕ ( g , ε ) = ϕ ( g , γε ) = ϕ (cid:0) ϕ ( g , γ ) , ε (cid:1) . (8 . . | g ε | = | ε | = | ϕ ( g , γ ) ε | , we deduce from (8.6.2) that g ε = ϕ ( g , γ ) ε , (8 . . α = α g γ, proving (i). In view of (8.6.3) and (8.6.4), point (ii) follows from (5.6).Conversely, assume (i–iv) and let us prove equality of the above germs. Setting δ = β γ , we have by (iv) that η := η = η ∈ Z ( δ ) , so it suffices to verify (8.6.1), which the reader could do without any difficulty. (cid:3) Proposition (8.6) then says that the typical situation in which an equality of germstakes place is (cid:2) α, g, β ; βγξ (cid:3) = (cid:2) αgγ, ϕ ( g, γ ) , βγ ; βγξ (cid:3) . Our next two results are designed to offer convenient representatives of germs.
Given any germ u , there exists an integer n , such that for every n ≥ n , (i) there is a representation of u of the form u = (cid:2) α , g , β ; β ξ (cid:3) , with | α | = n . (ii) there is a representation of u of the form u = (cid:2) α , g , β ; β ξ (cid:3) , with | β | = n . Proof.
Write u = (cid:2) α, g, β ; η (cid:3) , and choose any n ≥ max {| α | , | β |} . Then, for every n ≥ n we may write η = βγξ , with γ ∈ E ∗ , ξ ∈ E ∞ , and such that | γ | = n − | α | (resp. | γ | = n − | β | ). Therefore u = (cid:2) α, g, β ; βγξ (cid:3) = (cid:2) αgγ, ϕ ( g, γ ) , βγ ; βγξ (cid:3) , and we have | αgγ | = | α | + | gγ | = | α | + | γ | = n (resp. | βγ | = | β | + | γ | = n ). (cid:3) nverse semigroup actions and self-similar graphs 29 Given u and u in G tight ( S G,E ) , such that ( u , u ) ∈ G tight ( S G,E ) (2) ,that is, such that the multiplication u u is allowed or, equivalently, such that d ( u ) = r ( u ) , then there are representations of u and u of the form u = (cid:2) α , g , α ; α g ξ (cid:3) , and u = (cid:2) α , g , β ; βξ (cid:3) , and in this case u u = (cid:2) α , g g , β ; βξ (cid:3) . Proof.
Using (8.7), write u i = (cid:2) α i , g i , β i ; β i ξ i (cid:3) , with | β | = | α | . By virtue of ( u , u ) lying in G tight ( S G,E ) (2) , we have that β ξ = ( α , g , β )( β ξ ) = α g ξ , so in fact β = α , and ξ = g ξ . Then u = (cid:2) α , g , β ; β ξ (cid:3) = (cid:2) α , g , α ; α g ξ (cid:3) , and it suffices to put ξ = ξ , and β = β .With respect to the last assertion we have that u u = [ s ; βξ ], where s is the elementof S G,E given by s = ( α , g , α )( α , g , β ) (4.2) = ( α , g g , β ) , concluding the proof. (cid:3) Having extended the action of G to the set of infinite paths in Proposition (8.1), onemay ask whether it is possible to do the same for the cocycle ϕ . The following is anattempt at this which however produces a map taking values in the infinite product G ∞ ,rather than in G . We will denote by Φ, the mapΦ : G × E ∞ → G ∞ defined by the rule Φ( g, ξ ) n = ϕ ( g, ξ | n − ) , for g ∈ G , ξ ∈ E ∞ , and n ≥ G ∞ on the set { , , , . . . } , so the firstcoordinate of Φ( g, ξ ) is Φ( g, ξ ) = ϕ ( g, ξ | ) = ϕ (cid:0) g, r ( ξ ) (cid:1) (2.5.ii) = g. We wish to view Φ as some sort of cocycle but, unfortunately, property (2.5.x) doesnot hold quite as stated. On the fortunate side, a suitable modification of this relation,involving the left shift endomorphism λ of G ∞ , is satisfied: Let α be a finite path and let ξ be an infinite path such that d ( α ) = r ( ξ ) . Then, for every g in G , one has that Φ (cid:0) ϕ ( g, α ) , ξ (cid:1) = λ | α | (cid:0) Φ (cid:0) g, αξ ) (cid:1) . Proof.
For all n ≥
1, we haveΦ (cid:0) ϕ ( g, α ) , ξ (cid:1) n = ϕ (cid:0) ϕ ( g, α ) , ξ | n − (cid:1) = ϕ (cid:0) g, α ( ξ | n − ) (cid:1) == ϕ (cid:0) g, ( αξ ) | n − | α | (cid:1) = λ | α | (cid:0) Φ( g, αξ ) (cid:1) n . (cid:3) Another reason to think of Φ as a cocycle is as follows:
For every ξ ∈ E ∞ , and every g, h ∈ G , we have that Φ( gh, ξ ) = Φ (cid:0) g, hξ (cid:1) Φ( h, ξ ) . Proof.
We have for all n ∈ N , thatΦ( gh, ξ ) n = ϕ ( gh, ξ | n − ) (2.5.b) = ϕ (cid:0) g, h ( ξ | n − ) (cid:1) ϕ ( h, ξ | n − ) (8.1) == ϕ (cid:0) g, ( hξ ) | n − (cid:1) ϕ ( h, ξ | n − ) = Φ (cid:0) g, hξ (cid:1) n Φ( h, ξ ) n . (cid:3) The following elementary fact might perhaps justify the choice of “ n −
1” in thedefinition of Φ.
Given g ∈ G , and ξ ∈ E ∞ , one has that ( gξ ) n = Φ( g, ξ ) n ξ n . Proof.
By (8.1) we have that ( gξ ) | n = g ( ξ | n ), so the n th coordinate of gξ is also the n th coordinate of g ( ξ | n ). In addition we have that g ( ξ | n ) = g ( ξ | n − ξ n ) (2.5.ix) = g ( ξ | n − ) ϕ ( g, ξ | n − ) ξ n , so ( gξ ) n = ϕ ( g, ξ | n − ) ξ n = Φ( g, ξ ) n ξ n . (cid:3) We now wish to define a homomorphism (sometimes also called a one-cocycle) from G tight ( S G,E ) to the lag group ˘ G ⋊ ρ Z , by means of the rule (cid:2) α, g, β ; βξ (cid:3) (cid:16) ρ | α | (cid:0) Φ( g, ξ ) (cid:1) , | α | − | β | (cid:17) . As it is often the case for maps defined on groupoid of germs, the above tentativedefinition uses a representative of the germ, so some work is necessary to prove that thedefinition does not depend on the choice of representative. The technical part of this taskis the content of our next result. nverse semigroup actions and self-similar graphs 31
Suppose that ( G, E, ϕ ) is pseudo free. For each i = 1 , , let us be given ( α i , g i , β i ) in S G,E , as well as η i = β i ξ i ∈ Z ( β i ) . If (cid:2) α , g , β ; η (cid:3) = (cid:2) α , g , β ; η (cid:3) , then ρ | α | (cid:0) Φ( g , ξ ) (cid:1) ≡ ρ | α | (cid:0) Φ( g , ξ ) (cid:1) modulo G ( ∞ ) . Proof.
Assuming without loss of generality that | β | ≤ | β | , we may use (8.6) to write α = α g γ, g = ϕ ( g , γ ) , β = β γ , and η = η = β γξ, for suitable γ ∈ E ∗ and ξ ∈ E ∞ . Then necessarily ξ = γξ , and ξ = ξ , and ρ | α | (cid:0) Φ( g , ξ ) (cid:1) = ρ | α | + | γ | (cid:16) Φ (cid:0) ϕ ( g , γ ) , ξ (cid:1)(cid:17) (8.10) == ρ | α | ρ | γ | λ | γ | (cid:0) Φ (cid:0) g , γξ ) (cid:1) (7.3) ≡ ρ | α | (cid:0) Φ (cid:0) g , ξ ) (cid:1) . (cid:3)◮ Due to our reliance on Proposition (8.6) and Lemma (8.13), from now on and untilthe end of this section we will assume, in addition to (2.3), that (
G, E, ϕ ) is pseudo free.If g is in G ∞ , we will denote by ˘ g its class in the quotient group ˘ G . Likewise we willdenote by ˘Φ the composition of Φ with the quotient map from G ∞ to ˘ G . G × E ∞ −→ G ∞ −→ ˘ G Φ ............................................................................................................................................................................................................................................. ˘ΦIt then follows from (8.13) that the correspondence (cid:2) α, g, β ; βξ (cid:3) ∈ G tight ( S G,E ) ˘ ρ | α | (cid:0) ˘Φ( g, ξ ) (cid:1) ∈ ˘ G is a well defined map. This is an important part of the one-cocycle we are about tointroduce. The correspondence ℓ : (cid:2) α, g, β ; βξ (cid:3) (cid:16) ˘ ρ | α | (cid:0) ˘Φ( g, ξ ) (cid:1) , | α | − | β | (cid:17) gives a well defined map ℓ : G tight ( S G,E ) → ˘ G ⋊ ρ Z , which is moreover a one-cocycle. From now on ℓ will be called the lag function . Proof.
By the discussion above we have that the first coordinate of the above pair iswell defined. On the other hand, in the context of Proposition (8.6) one easily sees that | α | − | β | = | α | − | β | , so the second coordinate is also well defined.In order to show that ℓ is multiplicative, pick ( u , u ) ∈ G tight ( S G,E ) (2) . We may thenuse (8.8) to write u = (cid:2) α , g , α ; α g ξ (cid:3) , and u = (cid:2) α , g , β ; βξ (cid:3) . So ℓ ( u ) ℓ ( u ) = (cid:16) ρ | α | (cid:0) Φ( g , g ξ ) (cid:1) , | α | − | α | (cid:17)(cid:16) ρ | α | (cid:0) Φ( g , ξ ) (cid:1) , | α | − | β | (cid:17) == (cid:16) ρ | α | (cid:0) Φ( g , g ξ ) (cid:1) ρ | α | (cid:0) Φ( g , ξ ) (cid:1) , | α | − | α | + | α | − | β | (cid:17) == (cid:16) ρ | α | (cid:16) Φ( g , g ξ )Φ( g , ξ ) (cid:17) , | α | − | β | (cid:17) (8.11) = (cid:16) ρ | α | (cid:0) Φ( g g , ξ ) (cid:1) , | α | − | β | (cid:17) == ℓ (cid:0)(cid:2) α , g g , β ; βξ (cid:3)(cid:1) (8.8) = ℓ ( u u ) . (cid:3) The main relevance of this one-cocycle is that, together with the domain and rangemaps, it uniquely describes the elements of G tight ( S G,E ), as we will now show.
Given u , u ∈ G tight ( S G,E ) , one has that d ( u ) = d ( u ) r ( u ) = r ( u ) ℓ ( u ) = ℓ ( u ) ⇒ u = u . Proof.
Using (8.7), write u i = (cid:2) α i , g i , β i ; β i ξ i (cid:3) , for i = 1 ,
2, with | β | = | β | . Since β ξ = d ( u ) = d ( u ) = β ξ , we conclude that β = β , and ξ = ξ =: ξ. By focusing on the second coordinate of ℓ ( u i ), we see that | α | − | β | = | α | − | β | ,and hence | α | = | α | . Moreover, since α g ξ = α g ξ = r ( u ) = r ( u ) = α g ξ = α g ξ, we see that α = α , and g ξ = g ξ. (8 . . ℓ ( u ) = ℓ ( u ) also implies that˘ ρ | α | (cid:0) ˘Φ( g , ξ ) (cid:1) = ˘ ρ | α | (cid:0) ˘Φ( g , ξ ) (cid:1) , and since α = α , we conclude that ˘Φ( g , ξ ) = ˘Φ( g , ξ ) , and hence that there exists aninteger n such that ϕ ( g , ξ | n ) = ϕ ( g , ξ | n ) , ∀ n ≥ n . By (8.15.1) we also have that g ( ξ | n ) = g ( ξ | n ) , so (5.6) gives g = g , whence u = u . (cid:3) nverse semigroup actions and self-similar graphs 33 As a consequence of the above result we see that the map F : G tight ( S G,E ) → E ∞ × ( ˘ G ⋊ ˘ ρ Z ) × E ∞ defined by the rule F ( u ) = (cid:0) r ( u ) , ℓ ( u ) , d ( u ) (cid:1) , (8 . F has a natural groupoid structure, being the cartesianproduct of the lag group ˘ G ⋊ ˘ ρ Z by the graph of the transitive equivalence relation on E ∞ .Putting together (8.14) and (8.15) we may now easily prove: F is a groupoid homomorphism (functor), hence establishing an iso-morphism from G tight ( S G,E ) to the range of F . The range of F is then the concrete model of G tight ( S G,E ) we are after. But, beforegiving a detailed description of it, let us make a remark concerning notation: since theco-domain of F is a mixture of cartesian and semi-direct products, the standard notationfor its elements would be something like (cid:0) η, ( u, p ) , ζ (cid:1) , for η, ζ ∈ E ∞ , u ∈ ˘ G , and p ∈ Z .As part of our effort to avoid heavy notation we will instead denote such an element by (cid:0) η ; u, p ; ζ (cid:1) . The range of F is precisely the subset of E ∞ × ( ˘ G ⋊ ˘ ρ Z ) × E ∞ ,formed by the elements ( η ; ˘ g , p − q ; ζ ) , where η, ζ ∈ E ∞ , g ∈ G ∞ , and p, q ∈ N , are suchthat, for all n ≥ , (i) g n + p +1 = ϕ ( g n + p , ζ n + q ) , (ii) η n + p = g n + p ζ n + q . Proof.
Pick a general element (cid:2) α, g, β ; βξ (cid:3) ∈ G tight ( S G,E ) and, recalling that F ( (cid:2) α, g, β ; βξ (cid:3) ) = (cid:16) αgξ ; ˘ ρ | α | (cid:0) ˘Φ( g, ξ ) (cid:1) , | α | − | β | ; βξ (cid:17) , (8 . . η = αgξ, g = ρ | α | (cid:0) Φ( g, ξ ) (cid:1) , p = | α | , q = | β | , and ζ = βξ, so that the element depictedin (8.18.1) becomes ( η ; ˘ g , p − q ; ζ ), and we must now verify (i) and (ii). For all n ≥
1, onehas that g n + | α | = Φ( g, ξ ) n = ϕ ( g, ξ | n − ) , so η n + p = ( αgξ ) n + | α | = ( gξ ) n (8.12) = ϕ ( g, ξ | n − ) ξ n = g n + | α | ( βξ ) n + | β | = g n + p ζ n + q , proving (ii). Also, g n + p +1 = g n + | α | +1 = ϕ (cid:0) g, ξ | n (cid:1) = ϕ ( g, ξ | n − ξ n ) = ϕ (cid:0) ϕ ( g, ξ | n − ) , ξ n (cid:1) == ϕ (cid:0) g n + | α | , ( βξ ) n + | β | (cid:1) = ϕ (cid:0) g n + p , ζ n + q (cid:1) , proving (i) and hence showing that the range of F is a subset of the set described in thestatement.Conversely, pick η, ζ ∈ E ∞ , g ∈ G ∞ , and p, q ∈ N satisfying (i) and (ii), and let usshow that the element ( η ; ˘ g , p − q ; ζ ) lies in the range of F . Let g = g p +1 , α = η | p , and β = ζ | q , so ζ = βξ for a unique ξ ∈ E ∞ . We then claim that (cid:2) α, g, β ; βξ (cid:3) lies in G tight ( S G,E ). Inorder to see this notice that gd ( β ) = gd ( ζ q ) = gr ( ζ q +1 ) = r ( g p +1 ζ q +1 ) (ii) = r ( η p +1 ) = d ( η p ) = d ( α ) , so ( α, g, β ) ∈ S G,E , and therefore (cid:2) α, g, β ; βξ (cid:3) is indeed a member of G tight ( S G,E ). Theproof will then be concluded once we show that F ( (cid:2) α, g, β ; βξ (cid:3) ) = ( η ; ˘ g , p − q ; ζ ) , which in turn is equivalent to showing that(a) αgξ = η, (b) ˘ ρ | α | (cid:0) ˘Φ( g, ξ ) (cid:1) = ˘ g , ( c ) | α | − | β | = p − q, (d) βξ = ζ .Before proving these points we will show that ϕ ( g p +1 , ξ | n ) = g n + p +1 , ∀ n ≥ . ( † )This is obvious for n = 0. Assuming that n ≥ ϕ ( g p +1 , ξ | n ) = ϕ ( g p +1 , ξ | n − ξ n ) = ϕ (cid:0) ϕ ( g p +1 , ξ | n − ) , ξ n (cid:1) == ϕ ( g n + p , ζ n + q ) (i) = g n + p +1 , verifying ( † ).Addressing (a) we have to prove that ( αgξ ) k = η k , for all k ≥
1, but given that α isdefined to be η | p , this is trivially true for k ≤ p . On the other hand, for k = n + p , with n ≥
1, we have ( αgξ ) k = ( αgξ ) n + p = ( gξ ) n (8.12) = ϕ ( g, ξ | n − ) ξ n == ϕ ( g p +1 , ξ | n − ) ξ n ( † ) = g n + p ζ n + q (ii) = η n + p = η k , proving (a). Focusing on (b) we have for all n ≥ ρ | α | (cid:0) Φ( g, ξ ) (cid:1) p + n = Φ( g, ξ ) n = ϕ ( g p +1 , ξ | n − ) ( † ) = g n + p , proving that ρ | α | (cid:0) Φ( g, ξ ) (cid:1) ≡ g , modulo G ( ∞ ) , hence taking care of (b). The last twopoints, namely (c) and (d) are trivial and so the proof is concluded. (cid:3) nverse semigroup actions and self-similar graphs 35 As an immediate consequence we get a very precise description of the algebraic struc-ture of G tight ( S G,E ): Suppose that ( G, E, ϕ ) satisfies the conditions of (2.3) and is moreoverpseudo free. Then G tight ( S G,E ) is isomorphic to the sub-groupoid of E ∞ × ( ˘ G ⋊ ˘ ρ Z ) × E ∞ given by G G,E = ( η ; ˘ g , p − q ; ζ ) ∈ E ∞ × ( ˘ G ⋊ ˘ ρ Z ) × E ∞ : g ∈ G ∞ , p, q ∈ N , g n + p +1 = ϕ ( g n + p , ζ n + q ) ,η n + p = g n + p ζ n + q , for all n ≥ . Recall from [ ] that the C*-algebra of every graph is a groupoid C*-algebra for acertain groupoid constructed from the graph, and informally called the groupoid for the tail equivalence with lag .Viewed through the above perspective, our groupoid may also deserve such a denomi-nation, except that the lag is not just an integer as in [ ], but an element of the lag group˘ G ⋊ ρ Z precisely described by the lag function ℓ introduced in Proposition (8.14).
9. The topology of G tight ( S G,E ) . It is now time we look at the topological aspects of G tight ( S G,E ). In fact what we will dois simply transfer the topology of G tight ( S G,E ) over to G G,E via F . Not surprisingly F willturn out to be an isomorphism of topological groupoids.Recall from [ : Proposition 4.14] that, if S is an inverse semigroup acting on a locallycompact Hausdorff topological space X , then the corresponding groupoid of germs, say G ,is topologized by means of the basis consisting of sets of the formΘ( s, U ) , where s ∈ S , and U is an open subset of X , contained in the domain of the partialhomeomorphism attached to s by the given action. Each Θ( s, U ) is in turn defined byΘ( s, U ) = n [ s, x ] ∈ G : x ∈ U o . (9 . : 4.12] for more details.If we restrict the choice of the U ’s above to a predefined basis of open sets of X , e.g. thecollection of all cylinders in E ∞ in the present case, we evidently get the same topologyon the groupoid of germs. Therefore, referring to the model of G tight ( S G,E ) presented in(8.5), we see that a basis for its topology consists of the sets of the formΘ( α, g, β ; γ ) := n(cid:2) α, g, β ; ξ (cid:3) ∈ G tight ( S G,E ) : ξ ∈ Z ( γ ) o , (9 . α, g, β ) ∈ S G,E , and γ ∈ E ∗ . We may clearly suppose that | γ | ≥ | β | and, sinceΘ( α, g, β ; γ ) = ∅ , unless β is a prefix of γ , we may also assume that γ = βε , for some ε ∈ E ∗ . In this case, given any (cid:2) α, g, β ; ξ (cid:3) ∈ Θ( α, g, β ; γ ), notice that ξ ∈ Z ( γ ), and( α, g, β )( γ, , γ ) = (cid:0) αgε, ϕ ( g, ε ) , γ (cid:1) , from where one concludes that (cid:2) α, g, β ; ξ (cid:3) = (cid:2) αgε, ϕ ( g, ε ) , γ ; ξ (cid:3) , for all ξ ∈ Z ( γ ), and hence also thatΘ( α, g, β ; γ ) = Θ (cid:0) αgε, ϕ ( g, ε ) , γ ; γ (cid:1) . This shows that any set of the form (9.2) coincides with another such set for which β = γ . We may therefore do away with this repetition and redefineΘ( α, g, β ) := n(cid:2) α, g, β ; ξ (cid:3) ∈ G tight ( S G,E ) : ξ ∈ Z ( β ) o . (9 . The collection of all sets of the form Θ( α, g, β ) , where ( α, g, β ) rangein S G,E , is a basis for the topology of G tight ( S G,E ) . We may now give a precise description of the topology of G tight ( S G,E ), once it is viewedfrom the alternative point of view of Theorem (8.19):
For each ( α, g, β ) in S G,E , the image of Θ( α, g, β ) under F coincideswith the set Ω( α, g, β ) := ( η ; ˘ g , k ; ζ ) ∈ G G,E : η ∈ Z ( α ) , g ∈ G ∞ , k = | α | − | β | , ζ ∈ Z ( β ) , g | α | = g, g n + | α | +1 = ϕ ( g n + | α | , ζ n + | β | ) ,η n + | α | = g n + | α | ζ n + | β | , for all n ≥ , and hence the collection of all such sets form the basis for a topology on G G,E , with respectto which the latter is isomorphic to G tight ( S G,E ) as topological groupoids. Proof.
Left for the reader. (cid:3)
We may now summarize the main results obtained so far:
Suppose that ( G, E, ϕ ) satisfies the conditions of (2.3) and is moreoverpseudo free. Then O G,E is *-isomorphic to the C*-algebra of the groupoid G G,E describedin (8.19), once the latter is equipped with the topology generated by the basis of open sets Ω( α, g, β ) described in (9.3), for all ( α, g, β ) in S G,E . nverse semigroup actions and self-similar graphs 37 O G,E as a Cuntz-Pimsner algebra.
Inspired by Nekrashevych’s paper [ ], we will now give a description of O G,E as a Cuntz-Pimsner algebra [ ]. With this we will also be able to prove that O G,E is nuclear andthat G tight ( S G,E ) is amenable when G is an amenable group. As before, we will work underthe conditions of (2.3).We begin by introducing the algebra of coefficients over which the relevant Hilbertbimodule, also known as a correspondence, will later be constructed.Since the action of G on E preserves length by (2.4.iv), we see that the set of verticesof E is G -invariant, so we get an action of G on E by restriction. By dualization G actson the algebra C ( E ) of complex valued functions on E . We may therefore form thecrossed-product C*-algebra A = C ( E ) ⋊ G. Since C ( E ) is a unital algebra, there is a canonical unitary representation of G inthe crossed product, which we will denote by { v g } g ∈ G .On the other hand, C ( E ) is also canonically isomorphic to a subalgebra of A and wewill therefore identify these two algebras without further warnings.For each x in E , we will denote the characteristic function of the singleton { x } by q x , so that { q x : x ∈ E } is the canonical basis of C ( E ), and thus A coincides with theclosed linear span of the set (cid:8) q x v g : x ∈ E , g ∈ G (cid:9) . (10 . v g q x = q gx v g , ∀ x ∈ E , ∀ g ∈ G. (10 . A . In preparation for this wedenote by A e the right ideal of A generated by q d ( e ) , for each e ∈ E . In technical terms A e = q d ( e ) A. With the obvious right A -module structure, and the inner product defined by h y, z i = y ∗ z, ∀ y, z ∈ A e , one has that A e is a right Hilbert A -module. Notice that this is not necessarily a fullHilbert module since h A e , A e i is the two-sided ideal of A generated by q d ( e ) , which mightbe a proper ideal in some cases.As already seen in (10.1), A is spanned by the elements of the form q x v g . Therefore A e is spanned by the elements of the form q d ( e ) q x v g , but, since the q ’s are mutually orthogonal,this is either zero or equal to q d ( e ) v g . Therefore we see that A e = span { q d ( e ) v g : g ∈ G } . Notice that, since E is a finite set, C ( E ) is nothing but C | E | . Introducing the right Hilbert A -module which will later be given the structure of acorrespondence over A , we define M = M e ∈ E A e . Observe that if x is a vertex which is the source of many edges, say d − ( x ) = { e , e , . . . , e n } , then A e i = q d ( e i ) A = q x A, for all i , so that q x A appears many times as a direct summand of M . However these copiesof q x A should be suitably distinguished, according to which edge e i is being considered.On the other hand, notice that if d − ( x ) = ∅ , then q x A does not appear among thesummands of M , at all.Addressing the fullness of M , observe that h M, M i = X x ∈ E d − x ) = ∅ Aq x A, so, when E has no sinks , that is, when d − ( x ) is nonempty for every x , one has that M isfull. Given e ∈ E , the element q d ( e ) , when viewed as an element of A e ⊆ M , will play avery special role in what follows, so we will give it a special notation, namely t e := q d ( e ) . (10 . e , e ∈ E are such that x := d ( e ) = d ( e ) , then (10.3) assigns q x to both t e and t e . However the coordinate in which q x appears in t e i is determined by the corresponding e i , so if e = e , then t e = t e .In order to completely dispel any confusion, here is the technical definition: t e = ( m f ) f ∈ E , where m f = (cid:26) q d ( e ) , if f = e, , otherwise.We should notice that t e q d ( e ) = t e , (10 . y ∈ M may be written uniquely as y = X e ∈ E t e y e , (10 . y e ∈ A e .As the next step in constructing a correspondence over A , we would now like to definea certain *-homomorphism from A to the algebra L ( M ) of adjointable linear operators on M . Since A is a crossed product algebra, this will be accomplished once we produce acovariant representation ( ψ, V ) of the C*-dynamical system (cid:0) C ( E ) , G (cid:1) on M . We beginwith the group representation V . nverse semigroup actions and self-similar graphs 39 For each g ∈ G , let V g be the linear operator on M given by V g (cid:16) X e ∈ E t e y e (cid:17) = X e ∈ E t ge v ϕ ( g,e ) y e , whenever y e ∈ A e , for each e in E .By the uniqueness in (10.5), it is clear that V g is well defined. Each V g is a unitary operator on M . Moreover, the correspondence g V g is a unitary representation of G . Proof.
Pick g in G . We begin by claiming that the two sides in the expression defining V g , above, coincide whenever the y e are in A , even if y e does not belong to A e . Since V g is clearly additive, we only need to check that V g ( t e y ) = t ge v ϕ ( g,e ) y, ∀ y ∈ A. Observing that t e = t e q d ( e ) , we have V g ( t e y ) = V g ( t e q d ( e ) y ) = t ge v ϕ ( g,e ) q d ( e ) y == t ge q ϕ ( g,e ) d ( e ) v ϕ ( g,e ) y (2.5.vii) = t ge q d ( ge ) v ϕ ( g,e ) y = t ge v ϕ ( g,e ) y, proving the claim. One therefore concludes that V g is right- A -linear.We next claim that, for all e, f ∈ E , one has h V g ( t e ) , t f i = h t e , V g − ( t f ) i . (10 . . h V g ( t e ) , t f i = h t ge v ϕ ( g,e ) , t f i = v ∗ ϕ ( g,e ) h t ge , t f i = [ ge = f ] v − ϕ ( g,e ) q d ( ge ) == [ ge = f ] q ϕ ( g,e ) − d ( ge ) v − ϕ ( g,e ) (2.5.vii) = [ ge = f ] q d ( e ) v − ϕ ( g,e ) = ( ⋆ ) . Starting from the right-hand-side of (10.7.1), we have h t e , V g − ( t f ) i = h t e , t g − f v ϕ ( g − ,f ) i = [ e = g − f ] q d ( e ) v ϕ ( g − ,f ) == [ ge = f ] q d ( e ) v ϕ ( g,g − f ) − = [ ge = f ] q d ( e ) v − ϕ ( g,e ) , which agrees with ( ⋆ ) above, and hence proves claim (10.7.1). If y, z ∈ A , we then havethat h V g ( t e y ) , t f z i = y ∗ h V g ( t e ) , t f i z = y ∗ h t e , V g − ( t f ) i z = h t e y, V g − ( t f z ) i , from where one sees that h V g ( ξ ) , η i = h ξ, V g − ( η ) i , for all ξ, η ∈ M , hence proving that V g is an adjointable operator with V ∗ g = V g − . Let us next prove that V g V h = V gh , ∀ g, h ∈ G. By A -linearity it is enough to prove that these operators coincide on the set formed by the t e ’s, which is a generating set for M . We thus compute V g (cid:0) V h ( t e ) (cid:1) = V g (cid:0) t he v ϕ ( h,e ) (cid:1) = t ghe v ϕ ( g,he ) v ϕ ( h,e ) = t ghe v ϕ ( gh,e ) = V gh ( t e ) . Since it is evident that V is the identity operator on M we obtain, as a consequence,that V − g = V g − = V ∗ g , so each V g is unitary and the proof is concluded. (cid:3) In order to complete our covariant pair we must now construct a *-homomorphismfrom C ( E ) to L ( M ). With this in mind we give the following: For every x in E , let M x = M e ∈ r − ( x ) A e , which we view as a complemented sub-module of M . In addition, we let Q x be theorthogonal projection from M to M x , so that Q x ( t e y ) = [ r ( e )= x ] t e y, ∀ e ∈ E , ∀ y ∈ A. (10 . . Q x are pairwise orthogonal projections and that P x ∈ E Q x = 1 . Let ψ : C ( E ) → L ( M ) be the unique unital *-homomorphism suchthat ψ ( q x ) = Q x , ∀ x ∈ E . From our working hypothesis that E has no sources, we see that for every x in E ,there is some e ∈ E such that r ( e ) = x . So Q x ( t e ) = t e , whence Q x = 0. Consequently ψ is injective. The pair ( ψ, V ) is a covariant representation of the C*-dynamicalsystem (cid:0) C ( E ) , G (cid:1) in L ( M ) . Proof.
All we must do is check the covariance condition V g ψ ( y ) = ψ (cid:0) σ g ( y ) (cid:1) V g , ∀ g ∈ G, ∀ y ∈ C ( E ) , where σ is the name we temporarily give to the action of G on C ( E ). Since C ( E ) isspanned by the q x , it suffices to consider y = q x , in which case the above identity becomes V g Q x = Q gx V g . (10 . . M is generated, as an A -module, by the t e , for e ∈ E , so we only needto verify this on the t e . We have V g (cid:0) Q x ( t e ) (cid:1) = [ r ( e )= x ] V g ( t e ) = [ r ( e )= x ] t ge v ϕ ( g,e ) , while Q gx (cid:0) V g ( t e ) (cid:1) = Q gx (cid:0) t ge v ϕ ( g,e ) (cid:1) = [ r ( ge )= gx ] t ge v ϕ ( g,e ) , verifying (10.10.1) and concluding the proof. (cid:3) nverse semigroup actions and self-similar graphs 41 It follows from [ : Proposition 7.6.4 and Theorem 7.6.6] that there exists a *-homo-morphism Ψ : C ( E ) ⋊ G → L ( M ) , such that Ψ( q x ) = Q x , ∀ x ∈ E , and Ψ( v g ) = V g , ∀ g ∈ G. Equipped with the left- A -module structure provided by Ψ, we then have that M is acorrespondence over A .For later reference we record here a few useful calculations involving the left-modulestructure of M . Let g ∈ G , e ∈ E , and x ∈ E . Then (i) v g t e = t ge v ϕ ( g,e ) , (ii) q x v g t e = [ r ( ge )= x ] t ge v ϕ ( g,e ) . Proof.
We have v g t e = Ψ( v g ) t e = V g ( t e ) = t ge v ϕ ( g,e ) , proving (a). Also q x v g t e = Ψ( q x )( v g t e ) = Q x ( t ge v ϕ ( g,e ) ) = [ r ( ge )= x ] t ge v ϕ ( g,e ) . (cid:3) It is our next goal to prove that O G,E is naturally isomorphic to the Cuntz-Pimsneralgebra associated to the correspondence M , which we denote by O M . As a first step, weidentify a certain Cuntz-Krieger E -family. The following relations hold within O M . (a) For every x ∈ E , one has that P e ∈ r − ( x ) t e t ∗ e = q x . (b) P e ∈ E t e t ∗ e = 1 . ( c ) The set { q x : x ∈ E } ∪ { t e : e ∈ E } is a Cuntz-Krieger E -family. Proof.
We first claim that, for every x ∈ E , and every m ∈ M , one has that X e ∈ r − ( x ) t e t ∗ e m = q x m. To prove it, it is enough to consider the case in which m = t f , for f ∈ E , since thesegenerate M . In this case we have X e ∈ r − ( x ) t e t ∗ e t f = [ r ( f )= x ] t f t ∗ f t f = [ r ( f )= x ] t f (10.8.1) = Q x ( t f ) = q x t f , proving the claim. This says that the pair (cid:0) q x , P e ∈ r − ( x ) t e t ∗ e (cid:1) is a redundancy or, adoptingthe terminology of [ ], that the generalized compact operator X e ∈ r − ( x ) Ω t e ,t e is mapped to Ψ( q x ) via Ψ (1) . Therefore q x = X e ∈ r − ( x ) t e t ∗ e , in O M , proving (a). Point (b) then follows from the fact that P x ∈ E q x = 1 . Focusing now on (c), it is evident that { q x : x ∈ E } is a family of mutually orthogonalprojections. Moreover, for each e ∈ E , we have t ∗ e t e = h t e , t e i = q d ( e ) , proving (3.1.i) and also that t e is a partial isometry. Property (3.1.ii) also holds in viewof (a), so the proof is concluded. (cid:3) There exists a unique surjective *-homomorphism
Λ : O G,E → O M such that Λ( p x ) = q x , Λ( s e ) = t e , and Λ( u g ) = v g . Proof.
By the universal property of O G,E , in order to prove the existence of Λ it is enoughto check that the q x , t e , and v g satisfy the conditions of Definition (3.2).Condition (3.2.a) has already been proved above while (3.2.b) is evidently true since v is a representation of G in C ( E ) ⋊ G ⊆ O M . Condition (3.2.c) is precisely (10.11.i),while (3.2.d) was taken care of in (10.2).Since A is spanned by the q x and the v g by (10.1), and since M is generated over A by the t e , we see that O M is spanned by the set { q x , t e , v g : x ∈ E , e ∈ E , g ∈ G } , so Λ is surjective. (cid:3) Let us now prove that Λ is invertible by providing an inverse to it. Since A is thecrossed product C*-algebra C ( E ) ⋊ G , one sees that (3.2.a&d) guarantees the existenceof a *-homomorphism θ A : A → O G,E , sending the q x to the p x , and the v g to the u g . For each e in E , consider the linearmapping θ M : M → O G,E , nverse semigroup actions and self-similar graphs 43 given, for every m = ( m e ) e ∈ E ∈ M , by θ M ( m ) = X e ∈ E s e θ A ( m e ) ∈ O G,E . Notice that θ M ( t e ) = s e , for all e ∈ E , because θ M ( t e ) = s e θ A ( q d ( e ) ) = s e p d ( e ) = s e . The pair ( θ A , θ M ) is a representation of the correspondence M in thesense of [ : Theorem 3.4] , meaning that for all y ∈ A and all ξ, ξ ′ ∈ M , (i) θ M ( ξ ) θ A ( y ) = θ M ( ξy ) , (ii) θ A ( y ) θ M ( ξ ) = θ M ( yξ ) , (iii) θ M ( ξ ) ∗ θ M ( ξ ′ ) = θ A ( h ξ, ξ ′ i ) . Proof.
Considering the various spanning sets at our disposal, we may assume that y = q x v g ,that ξ = t e z , and ξ ′ = t e ′ z ′ , with x ∈ E , g ∈ G , e, e ′ ∈ E , z ∈ q d ( e ) A , and z ′ ∈ q d ( e ′ ) A .We then have θ M ( ξ ) θ A ( y ) = θ M ( t e z ) θ A ( y ) = s e θ A ( z ) θ A ( y ) = s e θ A ( zy ) = θ M ( t e zy ) = θ M ( ξy ) , proving (i). As for (ii), we have θ A ( y ) θ M ( ξ ) = θ A ( q x v g ) θ M ( t e z ) = p x u g s e θ A ( z ) = p x s ge u ϕ ( g,e ) θ A ( z ) == [ r ( ge )= x ] s ge θ A ( v ϕ ( g,e ) z ) = [ r ( ge )= x ] θ M ( t ge v ϕ ( g,e ) z (cid:1) (10.11.ii) = θ M ( q x v g t e z ) = θ M ( yξ ) , proving (ii). Focusing now on (iii), we have θ M ( ξ ) ∗ θ M ( ξ ′ ) = ( s e θ A ( z ) (cid:1) ∗ s e ′ θ A ( z ′ ) = [ e = e ′ ] θ A ( z ) ∗ p d ( e ) θ A ( z ′ ) == [ e = e ′ ] θ A ( z ∗ q d ( e ) z ′ ) = θ A ( h ξ, ξ ′ i ) . (cid:3) It is well known [ : Theorem 3.4] that the Toeplitz algebra for the correspondence M , usually denoted T M , is universal for representations of M , so there exists a *-homo-morphism Θ : T M → O G,E , coinciding with θ A on A and with θ M on M . The map Θ , defined above, factors through O M , providing a *-isomorphism Θ : O M → O G,E , such that Θ( q x ) = p x , Θ( t e ) = s e , and Θ( v g ) = u g , for all x ∈ E , e ∈ E , and g ∈ G . Proof.
The factorization property follows immediately from (10.12.b) and an easy modifi-cation of [ : Proposition 7.1] to Cuntz-Pimsner algebras.In order to prove that Θ is an isomorphism, observe that Θ ◦ Λ coincides with theidentity map on the generators of O G,E , by (10.13), and hence Θ ◦ Λ = id . The result thenfollows from the fact that Λ is surjective. (cid:3) If G is amenable then O G,E is nuclear.
Proof.
The amenability of G ensures that C ( E ) ⋊ G is nuclear. The result then followsfrom Theorem (10.15), the fact that Toeplitz-Pimsner algebras over nuclear coefficient al-gebras is nuclear [ : Theorem 4.6.25], and so are quotients of nuclear algebras [ : Theorem9.4.4]. (cid:3) Since E is finite, the nuclearity of C ( E ) ⋊ G is equivalent to theamenability of G . However, if the present construction is generalized for infinite graphs,one could produce examples of non amenable groups acting amenably on E , in which case C ( E ) ⋊ G would be nuclear. The proof of Corollary (10.16) could then be adapted toprove that O G,E is nuclear. If G is amenable, then G tight ( S G,E ) is an amenable groupoid. If more-over ( G, E, ϕ ) is pseudo free, then its sibling G G,E is an amenable groupoid.
Proof.
For G tight ( S G,E ), it follows from (10.16), (6.4) and [ : Theorem 5.6.18]. For G G,E ,it follows from (10.16), (9.6) and [ : Theorem 5.6.18]. (cid:3) Nekrashevych has proven in [ : Theorem 5.6], that a certain groupoid of germs,denoted D G , constructed in the context of self-similar groups, is amenable under the hy-pothesis that the group is contracting and self-replicating . Even though there are numerousdifferences between D G and G G,E , including a different notion of germs and Nekrashevych’srequirement that group actions be faithful , we believe it should be interesting to try togeneralize Nekrashevych’s result to our context.
11. Representing C ∗ ( E ) and G into O G,E . In this section we will study natural representations of the graph C*-algebra C ∗ ( E ) andof the group G in O G,E . As before, we keep (2.3) in force.Given that { p x : x ∈ E } ∪ { s e : e ∈ E } is a Cuntz-Krieger E -family, the universal property of the graph C*-algebra C ∗ ( E ) [ ]provides for the existence of a *-homomorphism ι : C ∗ ( E ) → O G,E , sending the canonical Cuntz-Krieger E -family of C ∗ ( E ) to the corresponding one within O G,E . The *-homomorphism ι above is injective. nverse semigroup actions and self-similar graphs 45 Proof.
Using the universal property of O G,E , it is easy to see that, for each complex number z , with | z | = 1, there is a *-homomorphism γ z : O G,E → O
G,E , satisfying γ z ( p x ) = p x , γ z ( s e ) = zs e , and γ z ( u g ) = u g , for all x ∈ E , e ∈ E and g ∈ G . It is also easy to see that the correspondence z → γ z defines an action of the circle group on O G,E , and moreover that ι is covariant relativeto this action on O G,E , on the one hand, and the standard gauge action on C ∗ ( E ), onthe other. In order to prove the injectivity of ι we may then apply the gauge invariantuniqueness Theorem [ : Theorem 2.2], which requires, in addition, that we verify thatthe p x are nonzero.To prove this we observe that, in the groupoid model of O G,E given by (9.6), for each x in E , the element p x is the characteristic function of the cylinder Z ( x ), seen as a subset of E ∞ , which in turn is the unit space of the groupoid G G,E . Since E has no sources, we havethat Z ( x ) is nonempty, whence p x is nonzero, as required. This concludes the proof. (cid:3) In particular, (11.1) implies that S E is ∗ -isomorphic to the inverse semigroup of O G,E generated by { s a : a ∈ E } .With respect to the injectivity of the representation of G into O G,E , we have to worka bit more to obtain a result in the line of (11.1).
Let π : S G,E → O
G,E and u : G → O G,E be the natural maps. If π isinjective, the so is u . Proof.
Let g ∈ G such that u g = u . For any x ∈ E we have π ( x, g, g − x ) = p x u g and π ( x, , x ) = p x u = p x . Since u g = u , we get ( x, g, g − x ) = ( x, , x ) ∈ S G,E , whence g = 1. (cid:3) We need to recall some extra definitions. Let G be an ´etale groupoid, i.e. a topologicalgroupoid whose unit space G (0) is locally compact and Hausdorff in the relative topology,and such that the range map r : G → G (0) is a local homeomorphism (and then so is thesource map d : G → G (0) ). An open set U ⊂ G is a slice if the restrictions of r and d to U are injective (see e.g. [ ]). In particular, G (0) is a slice [ : Proposition 3.4], and thecollection of all slices forms a basis for the topology of G [ : Proposition 3.5]. We denote by S ℓ ( G, E ) the set of all compact slices. It is well known(see e.g. [ : Proposition 2.2.4]) that S ℓ ( G, E ) forms a ∗ -inverse semigroup with the op-erations U V = { uv : u ∈ U, v ∈ V, ( u, v ) ∈ G (2) } , and U ∗ = { u − : u ∈ U } . Moreover, if U G,E = { U : U is a compact slice } ⊆ C ∗ ( G tight ( S G,E )) is the semigroupformed by their characteristic functions, then U G,E ∼ = S ℓ ( G, E ) . (11 . . Fix the canonical action θ of S G,E on E ∞ . Given any ( α, g, β ) ∈ S G,E , notice that thedomain Dom( θ ( α,g,β ) ) of the partial homeomorphism of E ∞ given by the action of ( α, g, β )is Z ( β ). Now, given ( α, g, β ) ∈ S G,E and any open set U ⊆ Z ( β ), set (see Section 9)Θ(( α, g, β ) , U ) = { [ α, g, β ; η ] : η ∈ U } . According to [ : Proposition 4.18], for every ( α, g, β ) ∈ S G,E and every open set U ⊆ Z ( β ),Θ(( α, g, β ) , U ) is a slice (in fact, they form a basis for the topology of G tight ( S G,E )). Then,we have the following result S ℓ ( G, E ) = h Θ(( α, g, β ) , Z ( β )) : ( α, g, β ) ∈ S G,E i . Proof.
By [ : Proposition 5.13(7)], C ∗ ( G tight ( S G,E )) is generated by { Θ(( α,g,β ) ,Z ( β )) : ( α, g, β ) ∈ S G,E } . Thus, the result holds by (11.3.1). (cid:3)
The next result is the key point for proving the injectivity of u : G → O G,E . If ( G, E, ϕ ) is pseudo free, then the map ρ : S G,E → S ℓ ( G, E )( α, g, β ) Θ(( α, g, β ) , Z ( β )) is a ∗ -semigroup isomorphism. Proof.
The surjectivity of ρ derives from (11.4).Now, let ( α, g, β ) , ( γ, h, η ) ∈ S G,E such that Θ(( α, g, β ) , Z ( β )) = Θ(( γ, h, η ) , Z ( η )).Then, for any ω ∈ Z ( β ) we have [ α, g, β ; ω ] = [ γ, h, η ; ω ]. Since ( G, E, ϕ ) is pseudofree, by (8.6) there exists τ ∈ E ∗ such that γ = α · gτ , η = βτ and h = ϕ ( g, τ ). If( α, g, β ) = ( γ, h, η ), then we can pick δ = τ in E ∗ and b ω = βδ e ω ∈ Z ( β ). Thus, b ω ∈ Z ( η )but [( γ, h, η ) , b ω ] is not defined, contradicting the hypothesis. Hence, ( α, g, β ) = ( γ, h, η ),whence ρ is injective, as desired. (cid:3) There exists a ∗ -isomorphism φ : O G,E → C ∗ ( G tight ( S G,E )) suchthat φ ( p x ) = 1 Θ(( x, ,x ) ,Z ( x )) for every x ∈ E , φ ( s a ) = 1 Θ(( a, ,d ( a )) ,Z ( a )) for every a ∈ E ,and φ ( u g ) = P x ∈ E Θ(( x,g,g − x ) ,Z ( g − x )) for every g ∈ G . Proof.
Notice that u g = P x ∈ E u g p x . Then it is direct but tedious to check that (cid:8) Θ(( x, ,x ) ,Z ( x )) : x ∈ E (cid:9) ∪ (cid:8) Θ(( a, ,d ( a )) ,Z ( a )) : a ∈ E (cid:9) ∪∪ ( X x ∈ E Θ(( x,g,g − x ) ,Z ( g − x )) : g ∈ G ) satisfy the defining relations for O G,E . Thus, by the Universal Property of O G,E , the map ϕ is an ∗ -homomorphism. Notice that ϕ is the homomorphism given in [ : Theorem 13.3],and so is injective. Surjectivity is due to [ : Proposition 5.13(7)]. (cid:3) nverse semigroup actions and self-similar graphs 47 If ( G, E, ϕ ) is pseudo free, then π : S G,E → O
G,E is injective.
Proof.
The composition map S G,E → O
G,E → C ∗ ( G tight ( S G,E ))( α, g, β ) s α u g s ∗ β Θ(( α,g,β ) ,Z ( β )) is injective by (11.4) and (11.5). By (11.6), O G,E → C ∗ ( G tight ( S G,E ))is injective. Thus, π is injective. (cid:3) Hence, we conclude If ( G, E, ϕ ) is pseudo free, then u : G → O G,E is injective.
Proof.
By Corollary (11.7), π is injective. Then, so is u by (11.2). (cid:3) In particular, (11.8) implies that, if ( G, E, ϕ ) is pseudo free, then S G,E is ∗ -isomorphic to the inverse semigroup of O G,E generated by { s a : a ∈ E } ∪ { p x u g : x ∈ E , g ∈ G } . Proposition (11.8) provides the best situation possible, as the next example shows:
Let E be the graph with only one vertex and one edge, and let G be any noncommutative group. Fix the trivial action of G on E , and let ϕ be the one-cocycle of G defined by ϕ ( g, a ) = 1 for every g ∈ G, a ∈ E . Then, it is easy to see that O G,E ∼ = C ∗ ( E ) ∼ = C ( T ), which is a commutative C*-algebra, so that it cannot contain anyfaithful copy of G .
12. The Hausdorff property for G tight ( S G,E ) . Again considering a triple (
G, E, ϕ ) satisfying (2.3), we will now give a characterization ofthe Hausdorff property for the tight groupoid of S G,E . The first result we may present inthis direction is: If ( G, E, ϕ ) is pseudo free, then G tight ( S G,E ) is a Hausdorff groupoid. Proof.
If (
G, E, ϕ ) is pseudo free, then S G,E is E*-unitary by (5.8), so G tight ( S G,E ) is Haus-dorff by [ : Corollary 3.17]. This could also be obtained from [ : Propositions 6.4 and6.2]. (cid:3) The converse of the above result is not true: as we will see in Example (18.15), thereare examples in which (
G, E, ϕ ) fails to be pseudo free but still G tight ( S G,E ) is Hausdorff.This may be interpreted as saying that the above assumption that (
G, E, ϕ ) is pseudofree is a much too strong hypothesis which one would therefore like to relax.On the other hand, recall from (5.5) that the failure of pseudo freeness for (
G, E, ϕ ) isequivalent to the existence of strongly fixed paths for nontrivial group elements. The resultbelow consists in allowing a limited amount of minimal strongly fixed paths, and hence alimited number of counter-examples for pseudo freeness, without harming Hausdorffnes.
Assuming that ( G, E, ϕ ) satisfies (2.3), the following are equivalent: (a) for every g in G , there are at most finitely many minimal strongly fixed paths for g , (b) G tight ( S G,E ) is Hausdorff. Proof.
We will of course use [ : Theorem 3.16]. So, given s in S G,E , we must provide afinite cover for J s . Since such a cover exists by trivial reasons when J s is empty, let usassume that s dominates at least one nonzero idempotent element. By (5.7) we then havethat s necessarily has the form s = ( α, g, α ) , and the set of nonzero idempotent elements dominated by s is given by J s = (cid:8) ( ατ, , ατ ) : τ ∈ E ∗ , d ( α ) = r ( τ ) , τ is strongly fixed by g (cid:9) . Using (5.3) we may further describe J s as J s = (cid:8) ( αµγ, , αµγ ) : µ ∈ M g , γ ∈ E ∗ , d ( α ) = r ( µ ) , d ( µ ) = r ( γ ) (cid:9) , (12 . . M g is the set of all minimal strongly fixed paths for g .Assuming (a), we have that M g is finite and then it is clear that (cid:8) ( αµ, , αµ ) : µ ∈ M g , d ( α ) = r ( µ ) (cid:9) is a finite cover for J s , whence G tight ( S G,E ) is Hausdorff by [ : Theorem 3.16].Assuming (b), let g ∈ G , and for each vertex x in E , denote by M xg the set of allminimal strongly fixed paths for g whose range coincides with x . Since E is finite, in orderto prove that M g is finite, it is enough to check that each M xg is finite.If M xg is empty, there is nothing to do, so let us assume the contrary. Given any µ in M xg , we then have that x = r ( µ ) = r ( gµ ) = gr ( µ ) = gx, so x is fixed by g . Consequently s := ( x, g, x ) lies in S G,E and, assuming (b), we have by[ : Theorem 3.16] that J s admits a finite cover which, in view of (12.2.1), must necessarilybe of the form (cid:8) ( µ i γ i , , µ i γ i ) } ni =1 , where the µ i ∈ M xg , and the γ i are paths with d ( µ i ) = r ( γ i ). We then claim that the µ i apearing above exhaust M xg , meaning that M xg = { µ i : i = 1 , . . . , n } . (12 . . µ ∈ M xg , so that ( µ, , µ ) ≤ s , by (5.7), and hence ( µ, , µ ) ∈ J s . Forsome i , one would then have that( µ i γ i , , µ i γ i )( µ, , µ ) = 0 , in which case either µ is a prefix of µ i γ i , or vice versa. This implies that either µ is a prefixof µ i , or vice versa, but since both µ and µ i are minimal, we must have µ = µ i , proving(12.2.2), and hence that M xg is finite. Consequently M g , which decomposes as the disjointunion of the M xg , is also finite. This verifies (a) and hence concludes the proof. (cid:3) nverse semigroup actions and self-similar graphs 49
13. Minimality for G tight ( S G,E ) . In this section we will study conditions under which G tight ( S G,E ) is minimal. Some of theresults we obtain here are analog to those proved in [ ] for the case of partial actions ofgroups.Given a triple ( G, E, ϕ ) satisfying (2.3), there are two relations among vertices in E which are relevant for the question at hand. First of all let us say that x ⇀ y provided there exists a path α in E ∗ such that d ( α ) = x and r ( α ) = y . Notice thatthis relation is reflexive (take α to be x ) and transitive (take the concatenation of therelevant paths). However this is neither symmetric nor antisymmetric, hence it is not anequivalence relation nor an order relation.The other relation we have in mind is simply the orbit relation, defined by x ∼ y when there exists g in G such that gx = y . Unlike “ ⇀ ”, it is well known that “ ∼ ” is anequivalence relation.We may then consider the smallest transitive relation extending both “ ⇀ ” and “ ∼ ”,by saying that vertices x and y are related when one may find a sequence of vertices x , x , . . . , x n such that x = x ⇀ x ∼ x ⇀ x ∼ . . . ∼ x n − ⇀ x n − ∼ x n = y. (13 . Let x and y be vertices in E . Then the following are equivalent; (i) there exists a vertex u such that x ⇀ u ∼ y , (ii) there exists a vertex v such that x ∼ v ⇀ y . Proof.
The fact that x ⇀ u ∼ y means that there exists a path α in E ∗ such that d ( α ) = x ,and r ( α ) = u , and there exists some g in G such that gu = y . Considering the path β = gα ,and the vertex v = gx , notice that d ( β ) = d ( gα ) = gd ( α ) = gx = v, while r ( β ) = r ( gα ) = gr ( α ) = gu = y, so x ∼ v ⇀ y . Conversely, assuming (ii) we have that gx = v = d ( β ) and r ( β ) = y , forsuitable g in G and β in E ∗ . Defining u = g − y , and α = g − β , we then have that d ( α ) = g − d ( β ) = x, and r ( α ) = g − r ( β ) = g − y = u, so x ⇀ u ∼ y . (cid:3) Given x and y in E , we will say that x ≫ y if the equivalent conditions of (13.2) are satisfied.Observe that “ ≫ ” coincides with the relation defined in (13.1), thanks to (13.2), andhence it is clearly transitive. It is also evident that “ ≫ ” is reflexive but, again, it isneither symmetric nor antisymmetric. Nevertheless we will view it as a defective orderrelation , in the sense that it satisfies all of the postulates of a (partial) order relation butfor antisymmetry.Anytime we have such a defective order relation, it is possible to turn it into a bonafide partial order by identifying elements whenever antisymmetry fails. By this we meanthat two vertices x and y in E will be called equivalent, in symbols x ≈ y whenever x ≫ y and y ≫ x . Writing [ x ] for the equivalence class of each x in E , the setof all equivalent classes, namely E ≈ = (cid:8) [ x ] : x ∈ E (cid:9) becomes a partially ordered set via the well defined order relation[ x ] ≥ [ y ] ⇐⇒ x ≫ y. Under the assumptions of (2.3), we will say that:(i) E is G -transitive if, for any two vertices x and y in E , one has that x ≫ y ,(ii) E is weakly G -transitive if, given any infinite path ξ , and any vertex x in E , there issome vertex v along ξ such that v ≫ x .The notion of G -transitivity generalizes the well known notion of transitivity in graphs.When it holds, E has a single equivalence class.On the other hand, weak G -transitivity is inspired by the notion of cofinality intro-duced in [ : Section 3], (see also [ : Definition 37.16]). The reader is however warned thatthe notions of weak G -transitivity and cofinality may only be reconciled upon a reversalof the direction of the edges in E , following the new trend in graph algebras started byKatsura (see the penultimate paragraph of the introduction in [ ]).It is evident that every G -transitive graph is weakly G -transitive, but these are some-times equivalent notions as we will now show: In addition to the assumptions in (2.3), suppose that E has no sinks ,meaning that d − ( x ) is nonempty for every x in E . Then, if E is weakly G -transitive, itmust also be G -transitive. nverse semigroup actions and self-similar graphs 51 Proof.
Since E is finite, we may choose a minimal element [ x ] in E / ≈ . Using that E has no sinks, we may find an infinite sequence of edges . . . , α − i − , α − i , . . . , α − , α − , α ∈ E , such that d ( α ) = x , and d ( α i − ) = r ( α i ), for every i ≤
0. Since E is also finite, theremust be repetitions among the α i , say α m = α n , where m < n ≤
0. The finite path γ = α m α m +1 . . . α n − , therefore satisfies d ( γ ) = d ( α n − ) = r ( α n ) = r ( α m ) = r ( γ ) , and hence γ may be concatenated with itself infinitely many times producing the infinitepath ξ = γγγ . . . Given any y in E , and assuming weak G -transitivity, there is some vertex v along ξ ,such that v ≫ y . Since ξ is made of repetitions of γ , one has that v = r ( α k ), for some k in the integer interval [ m, n ]. We then have x = d ( α ) = d ( α k . . . α − α − α ) ⇀ r ( α k ) = v ≫ y, so x ≫ y , but since [ x ] is minimal, we deduce that [ x ] = [ y ], which is to say that x ≈ y .The conclusion is that E / ≈ is a singleton, from where G -transitivity follows. (cid:3) Of course the above result has taken advantage of the fact that E is a finite graph inan essential way, so nothing like this is to be expected for infinite graphs.Regardless of the absence of sinks, we have: Given ( G, E, ϕ ) satisfying (2.3), one has that the following are equiva-lent: (i) the standard action of S G,E on E ∞ defined in (8.3) is irreducible, (ii) G tight ( S G,E ) is minimal, (iii) E is weakly G -transitive. Proof.
The equivalence between (i) and (ii) is a consequence of [ : Proposition 5.4]. Wewill next show that the above condition (iii) is equivalent to condition (iii) of [ : Theorem5.5], from where the result will follow. In doing so, it is useful to understand how doidempotents in E behave under conjugation by elements in S G,E , and we leave it for thereader to verify that, given ( α, g, β ) in S G,E and ( γ, , γ ) ∈ E , one has that( α, g, β )( γ, , γ )( α, g, β ) ∗ = ( αgε, , αgε ) , if γ = βε, ( α, , α ) , if γε = β, , otherwise . (13 . . (iii) ⇒ [ : Theorem 5.5.iii]: Given any two nonzero idempotent elements in E , necessarilyof the form f α = ( α, , α ), and f β = ( β, , β ) , to employ the notation introduced in (4.5), we must find an outer cover of f α (in the senseof [ : Definition 2.9]) formed by a finite number of conjugates of f β . As a first step, noticethat s := (cid:0) d ( β ) , , β (cid:1) lies in S G,E and sf β s ∗ = (cid:0) d ( β ) , , β (cid:1) ( β, , β ) (cid:0) β, , d ( β ) (cid:1) = (cid:0) d ( β ) , , d ( β ) (cid:1) = f d ( β ) . Thus, anything that may be obtained by conjugating f d ( β ) by an element t ∈ S G,E , mayalso be obtained by conjugating f β by ts . It therefore suffices to find an outer cover of f α formed by conjugates of f d ( β ) .On the other hand, observe that f α ≤ f r ( α ) , so any outer cover of f r ( α ) is necessarilyalso an outer cover of f α . This said we see that we may assume, without loss of generality,that α and β are vertices.Our task thus gets simplifyed in the sense that we now need to find an outer cover of f x made of conjugates of f y , for any given vertices x and y in E .Recall from (8.2) that the set of all infinite paths with a given prefix γ is denoted Z ( γ ). In case we take γ = x , we then have that Z ( x ) is the set of all infinite paths withrange x .Thanks to weak G -transitivity, for each ξ in Z ( x ), we may choose a vertex v ξ along ξ such that v ξ ≫ y . This is to say that we may write ξ = α ξ η ξ , where α ξ is a finite path, η ξ is an infinite path and d ( α ξ ) = v ξ . • • yu ξ ................................................................................................................................................................................................................................................................... .............. β ξ • v ξ • x ............................................................................................................................................................................................................................................................................ .............. η ξ ................................................................................................................................................................................................................................................................... .............. α ξ .............................. g ξ .............................................................................. | {z } ξ The fact that v ξ ≫ y may be expressed by saying that v ξ ∼ u ξ ⇀ y , for some vertex u ξ , so there exists g ξ in G , and a finite path β ξ , such that g ξ u ξ = v ξ , d ( β ξ ) = u ξ , and r ( β ξ ) = y .Speaking of the cylinders Z ( α ξ ), it is obvious that ξ ∈ Z ( α ξ ), so we see that thecollection of cylinders (cid:8) Z ( α ξ ) (cid:9) ξ ∈ Z ( x ) is an open cover (in the topological sense of the word) for Z ( x ). Since Z ( x ) is compact,we may extract a finite subcover, say Z ( x ) ⊆ S ξ ∈ F Z ( α ξ ) , (13 . . nverse semigroup actions and self-similar graphs 53 where F is a finite subset of Z ( x ). We next claim that { f α ξ } ξ ∈ F is an outer cover of f x .To see this, let e be a nonzero idempotent in E , with e ≤ f x . Then e is necessarilygiven by e = ( γ, , γ ), for some finite path γ such that r ( γ ) = x . Using our standinghypothesis (2.3) according to which E has no sources, we may prolong γ to an infinitepath η , which will then share ranges with γ , whence η ∈ Z ( x ). By (13.6.2) we then havethat η lies in Z ( α ξ ), for some ξ ∈ F .This implies that both α ξ and γ are prefixes of η , from where it is easy to see that α ξ is a prefix of γ or vice-versa. In particular we conclude that f α ξ ⋓ f γ , proving our claim.Incidentally this could also be obtained from [ : Proposition 3.8].We next claim that each f α ξ is a conjugate of f y . To see this, observe that, since g ξ d ( β ξ ) = g ξ u ξ = v ξ = d ( α ξ ) , one has that s ξ := ( α ξ , g ξ , β ξ ) lies in S G,E , and s ξ f y s ∗ ξ = ( α ξ , g ξ , β ξ )( y, , y )( β ξ , g ξ , α ξ ) = ( α ξ , , α ξ ) = f α ξ . This concludes the proof of condition (iii) of [ : Theorem 5.5].[ : Theorem 5.5.iii] ⇒ (iii): Given any infinite path ξ , and any vertex y in E , we mustshow that there is some vertex v along ξ such that v ≫ y . Letting x = r ( ξ ), let us use thehypothesis regarding the nonzero idempotents f x = ( x, , x ), and f y = ( y, , y ) . This is to say that there are s , s , . . . , s n in S G,E , such that { s i f y s ∗ i } ≤ i ≤ n is an outercover for f x . For each i , write s i = ( α i , g i , β i ), so that s i f y s ∗ i = ( α i , g i , β i )( y, , y )( β i , g i , α i ) . Observe that, unless r ( β i ) = y , the element displayed above vanishes, so it cannotpossibly have any use as a member of a cover. We may then safely discard it, being leftonly with those β i such that that r ( β i ) = y . In this case, by the second clause in (13.6.1)we have s i f y s ∗ i = ( α i , , α i ) = f α i . Unless r ( α i ) = x , notice that f α i ⊥ f x , in which case f α i may again be discarded as itplays no role in an outer cover for f x . We may therefore suppose, without loss of generalitythat r ( α i ) = x , for all i . • • y ................................................................................................................................................................................................................................................................... .............. β i • • x ................................................................................................................................................................................................................................................................... .............. α i .............................. g i .............................................................................. This is in fact a cover but we do not need to worry about this right now.
Given that ( α i , g, β i ) lies in S G,E , we necessarily have that d ( α i ) = g i d ( β i ). Recallingthat the infinite path ξ , chosen at the beginning of the present argument, has range x , weclaim that ξ is necessarily of the form ξ = α i ξ ′ , for some i and some infinite path ξ ′ . To see this, write ξ = δξ ′′ , where ξ ′′ is an infinite path and δ is a finite path whose length exceeds the length of all ofthe α i . Observing that r ( δ ) = x , we then have that f δ := ( δ, , δ ) ≤ ( x, , x ) = f x . So, by the covering property we must have f δ ⋓ f α i , for some i, which implies that either δ is a prefix of α i or vice versa. However, due to the fact that | δ | > | α i | , by construction,the first alternative cannot hold, meaning that α i is a prefix of δ , and hence also of ξ ,proving the claim.It follows that d ( α i ) is a vertex along ξ , and it is clear from the above diagram that d ( α i ) ≫ y . This concludes the proof. (cid:3) Combining the above result with (13.5), we immediately deduce:
If, in addition to the assumptions of (13.6) we have that E has no sinks,then conditions (13.6.i–iii) are also equivalent to: (iv) E is G -transitive. It is interesting to observe that G -transitivity, when it holds, is the result of a jointeffort by the action of G and the edges, both of which may be seen as pushing verticesaround. However, sometimes only one of the players bear the responsibility to do thepushing around:(1) If G acts transitively on E , then E is G -transitive regardless of the graph. Easyexamples of this situation may be built on a graph formed by a disjoint union ofloops, for instance.(2) If G fixes all vertices, then E is (weakly) G -transitive if and only if E is (weakly)transitive [ : Definition 37.16]. This is the case of Katsura algebras, when seen in thepresent framework.
14. Essentially principal groupoids.
In this section we will discuss conditions under which G tight ( S G,E ) is an essentially prin-cipal groupoid, a condition which is intimately tied to the action of S G,E on E ∞ beingtopologically free. The reader is referred to [ : Section 4] for the definition of the notionof topologically free actions of inverse semigroups, as well as some of the main tools weshall use here. nverse semigroup actions and self-similar graphs 55 (1) A circuit is a finite path γ ∈ E ∗ of nonzero length such that d ( γ ) = r ( γ ).(2) A G -circuit is a pair ( g, γ ), where g ∈ G , and γ ∈ E ∗ is a finite path of nonzero lengthsuch that d ( γ ) = gr ( γ ). • .......................................................................................................................................................................................... γ • .................................................................................................................................................... γ • ..................................................................................................................................................... γ • ..................................................................................... ........................................................................ γ • ...................................................................................... ............ ............................................................. γ • ..................................................................................................................... ............................................................................. γ • .......................................... g A G -circuit. Thus, a G -circuit needs a little help from the group to close it up. Notice that a (usual)circuit γ may be concatenated with itself infinitely many times producing an infinite path ξ = γγγ . . . Moreover, if s = (cid:0) γ, , d ( γ ) (cid:1) , then, regarding the standard action of S G,E on E ∞ defined in (8.3), it is easy to see that sξ = ξ, which is to say that ξ is a fixed point for s . It is also possible to create fixed pointsfrom G -circuits as follows: Given a G -circuit ( g, γ ) , define a sequence { γ n } n ≥ of finite paths,and a sequence { g n } n ≥ of group elements, recursively by γ = γ , g = g , and ( γ n +1 = g n γ n g n +1 = ϕ ( g n , γ n ) , for all n ≥ . Then (i) d ( γ n ) = r ( γ n +1 ) , for all n ≥ , (ii) the concatenation ξ = γ γ γ . . . is a well defined infinite path, (iii) for every finite path β such that d ( β ) = r ( γ ) , one has that s := ( βγ, g, β ) lies in S G,E ,and βξ is a fixed point for s . Circuits are also called loops or cycles in the graph C*-algebra literature. Our preference for circuits comes from the fact that it is the terminology of choice among graph theorists and also because in theestablished graph theory terminology the word loop refers to a single edge whose source and range coincide.
Proof.
In order to prove the case n = 1 of (i), we have d ( γ ) = d ( γ ) = gr ( γ ) = r ( gγ ) = r ( g γ ) = r ( γ ) . For n ≥
1, we have d ( γ n +1 ) = d ( g n γ n ) = g n d ( γ n ) (2.5.vii) = ϕ ( g n , γ n ) d ( γ n ) = g n +1 d ( γ n ) ( ⋆ ) == g n +1 r ( γ n +1 ) = r ( g n +1 γ n +1 ) = r ( γ n +2 ) , where we have used induction in the step marked with ( ⋆ ) above. This proves (i), which inturn immediately implies (ii).In order to show that the element s defined in (iii) indeed lies in S G,E , it is enoughto observe that gd ( β ) = gr ( γ ) = d ( γ ) = d ( βγ ) . Before proving the last part of (iii), we claim that g n ( γ n γ n +1 . . . γ n + k ) = γ n +1 γ n +2 . . . γ n + k +1 , ∀ n ≥ , ∀ k ≥ . In case k = 0, this is true by the recursive definition above, and if k ≥
1, we have g n ( γ n γ n +1 . . . γ n + k ) = ( g n γ n ) ϕ ( g n , γ n )( γ n +1 . . . γ n + k ) == γ n +1 g n +1 ( γ n +1 . . . γ n + k ) , and the claim then follows easily by induction. A useful consequence is that g ( γ γ . . . γ n ) = γ γ . . . γ n +1 , ∀ n ≥ , from where we further deduce that gξ = g ( γ γ γ . . . ) = γ γ γ . . . (14 . . s ( βξ ) = ( βγ, g, β )( βξ ) (8.3) = βγgξ (14.2.1) = βγγ γ . . . γ n +1 = βξ. (cid:3) The above method does not give us all fixed points of every single element s in S G,E ,but in certain cases it does:
Given s := ( α, g, β ) in S G,E , suppose that | α | > | β | . Then, regardingthe standard action of S G,E on E ∞ , one has that: (i) s admits at most one fixed point, (ii) if s admits a fixed point ζ , then there is a G -circuit ( g, γ ) such that α = βγ , and ζ coincides with the fixed point βξ mentioned in (14.2.iii), constructed from ( g, γ ) . nverse semigroup actions and self-similar graphs 57 Proof.
Assuming that ζ is a fixed point for s , it must lie in Z ( β ), so necessarily ζ = βξ ,for a suitable infinite path ξ . We then have βξ = ζ = sζ = ( α, g, β )( βξ ) = αgξ. (14 . . α and β are prefixes of ζ , so one must be a prefix of the other,but since | α | > | β | , the only alternative is that β is a prefix of α . We may therefore write α = βγ, for some finite path γ , which necessarily satisfies gr ( γ ) = gd ( β ) = d ( α ) = d ( γ ) . In other words, ( g, γ ) is a G -circuit. From (14.3.1) we also deduce that βξ = αgξ = βγgξ, so ξ = γgξ. Let us now write ξ = γ γ γ . . . , where each γ i is a finite path with | γ i | = | γ | .Then γ γ γ . . . = ξ = γgξ = γg ( γ γ γ . . . ) = γ ( g γ )( g γ )( g γ ) . . . , where the g i are recursively defined by g = g , and g n +1 = ϕ ( g n , γ n ). It then follows that γ = γ , and γ n +1 = g n γ n , for all n ≥
1, so we see that the γ n and the g n are preciselydefined as in (14.2). This concludes the proof. (cid:3) As already announced we will eventually be interested in determining conditions underwhich the standard action of S G,E on E ∞ is topologically free, so the fixed points thatwill really interest us are the interior ones.Under the conditions of the above result, when there is at most one fixed point, theexistence of interior fixed points hinges on whether or not the unique fixed point is isolatedin E ∞ . We will now introduce certain concepts designed to study isolated fixed points.Recall from (2.3) that our graph E has no sources, meaning that r − ( x ) is nonemptyfor every vertex x . (1) We shall say that a vertex x in E is a simple vertex if r − ( x ) is a singleton.(2) Given a path γ = γ γ . . . γ n in E ∗ , where each γ i is in E , we will say that γ has noentry if d ( γ i ) is a simple vertex for every i = 1 , . . . , n .(3) If the condition above fails, we will say that γ has an entry .Thus, if a path γ = γ γ . . . γ n has no entry, then r − (cid:0) d ( γ i ) (cid:1) is a singleton for every i , and we may obviously guess which is the edge forming this singleton, namely r − (cid:0) d ( γ i ) (cid:1) = { γ i +1 } , provided i < n . However the same cannot be said when i = n , unless ( g, γ ) is a G -circuit,in which case r − (cid:0) d ( γ n ) (cid:1) = { gγ } . The notion of entryless paths will only be useful when applied to G -circuits. Under the conditions of (14.3.ii), let ( g, γ ) be the G -circuit and ζ bethe fixed point for s mentioned there. Then the following are equivalent: (i) ζ is an isolated point in E ∞ , (ii) γ has no entry. Proof.
In case γ has no entry, writing γ = γ γ . . . γ n , where the γ i are edges, notice thatthe only infinite path extending γ is the path ξ referred to in (14.2.ii). The fixed point ζ = βξ mentioned in (14.3.ii) is therefore the only infinite path extending βγ , whence Z ( βγ ) = { ζ } , which implies that ζ is isolated.Conversely, assuming that ζ is isolated, there exists a sufficiently long prefix ε of ζ ,such that Z ( ε ) = { ζ } . This means that ζ is the only infinite path extending ε . Writing ζ = ζ ζ ζ . . . , where the ζ i are edges, one then has that, for sufficiently large i , there is only one edgewhose range is d ( ζ i ). Letting { γ n } n ≥ and { g n } n ≥ be the sequences defined in (14.2), wethen have that, for sufficiently large n , the G -circuit ( g n , γ n ) has no entry. Since G acts on E by graph automorphisms, we may easily prove by induction that all G -circuits ( g k , γ k )have no entry, including ( g , γ ) = ( g, γ ). (cid:3) Since we are interested in topologically free actions, we would like to avoid isolatedfixed points and hence we will be interested in situations when every G -circuit has anentry. However, given that we are working with finite graphs only, the action of G on E turns out not to be relevant in this respect. In precise terms, what we mean is that: Under the conditions of (2.3), the following are equivalent: (i) every G -circuit has an entry, (ii) every circuit has an entry. Proof.
Since every circuit γ gives rise to the G -circuit (1 , γ ), it is evident that (i) implies(ii). Conversely, assume (ii) and let γ be a G -circuit. Leting { γ n } n ≥ and { g n } n ≥ be asin (14.2), consider the infinite path ξ = γ γ γ . . . mentioned in (14.2.iii). Notice that the γ i are all in the orbit of γ under the action of G , and hence the length of γ i coincides withthat of γ . As E is finite, there is only a finite number of paths of this length, so there mustnecessarily be repetitions among the γ i , say γ i = γ j , where i < j . Then r ( γ i +1 ) = d ( γ i ) = d ( γ j ) , so the path γ i +1 γ i +2 . . . γ j is a circuit, which by hypothesis has an entry. It is now easy to see that some γ k musthave an entry. Finally, since γ k is in the orbit of γ under the action of G , then γ likewisehas an entry, concluding the proof. (cid:3) nverse semigroup actions and self-similar graphs 59 Observe that we have used the finiteness of E in a very strong way above. Thus, shouldour theory ever be extended to infinite graphs, one might have to distinguish betweenconditions (14.6.i) and (14.6.ii).We should point out that a graph in which every circuit has an entry is usually saidto satisfy condition (L).The above results, mainly (14.3) and (14.5), may also be used to study the fixed pointsfor elements s := ( α, g, β ) when | α | < | β | , since such fixed points are precisely the sameas the fixed points of s ∗ , and s ∗ clearly satisfies the hypothesis of (14.3). However we stillhave work to do in order to treat the remaining case | α | = | β | . Let s := ( α, g, β ) ∈ S G,E , whith | α | = | β | , and suppose that s admitsa fixed point. Then (i) α = β , (ii) the fixed points of s in E ∞ are precisely the elements of the form ζ = βξ , where ξ isan infinite path such that r ( ξ ) = d ( β ) , and gξ = ξ . Proof.
Left for the reader. (cid:3)
The conclusion of the previous Proposition is that when | α | = | β | , understanding thefixed points for s requires understanding the fixed points for the action of g on E ∞ . Onemay easily describe such fixed points in terms of the action of G on E and the cocycle ϕ ,but apparently there is no smart way to control each and every one of them. However,since our main interest is in studying topological freeness, we need only focus on large(meaning open) sets of fixed points: Suppose that s := ( α, g, α ) lies in S G,E , and that ζ is an interiorfixed point for s . Then there is a finite path γ , such that: (i) gγ = γ , (ii) d ( α ) = r ( γ ) , (iii) ζ ∈ Z ( αγ ) , (iv) the group element h := ϕ ( g, γ ) pointwise fixes the cylinder Z ( d ( γ )) .Conversely, if γ is any finite path satisfying (i), (ii) and (iv), then every ζ ∈ Z ( αγ ) is a(necessarily interior) fixed point for s . Proof.
In particular ζ a fixed point for s so, by (14.7) we have that ζ = αξ , with gξ = ξ .Moreover there exists a neighborhood U of ζ consisting of fixed points for ζ . Since thecylinders form a basis for the topology of E ∞ , we may assume without loss of generalitythat U = Z ( β ), for some finite path β , which we may assume is as long as we wish, andour wish in this case is simply that | β | > | α | .Since ζ lies in Z ( β ), we have that β is a prefix of ζ , so we may write ζ = βη , for someinfinite path η . We then have βη = ζ = αξ. By this we mean that every point in Z ( d ( γ )) is fixed by h . Given that | β | > | α | , this implies that α is a prefix of β , so we write β = αγ , for asuitable finite path γ , obviously satisfying (ii). Consequently ζ = βη = αγη ∈ Z ( αγ ) , proving (iii). Given any infinite path µ ∈ Z ( d ( γ )), we may form the path αγµ , whichnecessarily lies in Z ( αγ ) = Z ( β ), and hence is fixed under s . Therefore αγµ = ( α, g, α )( αγµ ) = αg ( γµ ) = α ( gγ ) (cid:0) ϕ ( g, γ ) µ (cid:1) = α ( gγ ) ( hµ ) , whence γ = gγ , proving (i), and µ = hµ , in turn proving (iv).In order to prove the last sentence in the statement it is enough to notice that any el-ement in Z ( αγ ) is necessarily of the form αγµ , where µ ∈ Z ( d ( γ )), and the last calculationdisplayed above could be used to check that αγµ is fixed under s . (cid:3) Searching for conditions under which the standard action of S G,E on E ∞ is topo-logically free, one should probably worry about group elements fixing whole cylinders, asin (14.8.iv). The following notion is designed to pinpoint situations under which wholecylinders of the form Z ( x ) are in fact fixed. Given g ∈ G , and x ∈ E , we shall say that g is slack at x , if thereis a non-negative integer n such that all finite paths γ with r ( γ ) = x , and | γ | ≥ n , arestrongly fixed by g , as defined in (5.1).As already discussed at the begining of section (5), if γ is strongly fixed by g , then g fixes any finite path extending γ , and hence also all infinite paths in Z ( γ ).If g is slack at x , and if n is as in (14.9), notice that Z ( x ) = S r ( γ )= x | γ | = n Z ( γ ) , and since each γ occuring above is strongly fixed by g , we have that g pointwise fixes Z ( γ ),and hence also the whole cylinder Z ( x ).Notice that a path of length zero, namely a vertex x , is never strongly fixed by anontrivial group element g , because ϕ ( g, x ) (2.5.ii) = g = 1 . The concept of slackness above should therefore be seen as the best replacement forthe notion of being strongly fixed in case of a vertex.We are now ready for a main result:
Under the conditions of (2.3), the standard action of S G,E on E ∞ istopologically free if and only if the following two conditions hold: (i) every G -circuit has an entry , (ii) given a vertex x , and a group element g fixing every infinite path in Z ( x ) , thennecessarily g is slack at x . Recall that this is the same as saying that every circuit has an entry by (14.6). nverse semigroup actions and self-similar graphs 61
Proof.
Suppose (i) and (ii) hold and let ζ be an interior fixed point for some s = ( α, g, β )in S G,E . In order to prove topological freeness, we need to prove that ζ is a trivial fixedpoint for s . Case 1:
Let us first assume that | α | = | β | . Letting γ and h as in (14.8), we then have that h pointwise fixes the cylinder Z ( d ( γ )). By (ii) we then conclude that h is slack at d ( γ ), sothere is n such that every finite path of length n and range d ( γ ) is strongly fixed by h .By (14.8.iii) we have that ζ lies in Z ( αγ ), so we may write ζ = αγξ , for some infinitepath ξ with d ( γ ) = r ( ξ ). Denoting by ε the path formed by the first n edges of ξ , we thenhave that r ( ε ) = r ( ξ ) = d ( γ ) , so ε is strongly fixed by h , and we may further write ζ = αγεξ ′ , for a suitable infinite path ξ ′ . If follows that ζ ∈ Z ( αγε ), which is the domain of theidempotent f αγε = ( αγε, , αγε ) . In addition sf αγε = ( α, g, α )( αγε, , αγε ) = (cid:0) αg ( γε ) , ϕ ( g, γε ) , αγε (cid:1) , (14 . . f αγε . Tosee this notice that g ( γε ) = ( gγ ) (cid:0) ϕ ( g, γ ) ε (cid:1) (14.8.i) = γhε = γε, while ϕ ( g, γε ) (2.5.x) = ϕ (cid:0) ϕ ( g, γ ) , ε (cid:1) = ϕ (cid:0) h, ε (cid:1) = 1 . Plugging the last two identities at the end of (14.10.1) leads to sf αγε = f αγε , thusproving that ζ is a trivial fixed point, as needed. Case 2:
Let us now assume that | α | > | β | . By (14.3) we have that ζ is the only fixedpoint for s , necessarily given in terms of a G -circuit ( g, γ ), as in (14.3.iii).Being the unique fixed point, as well as an interior member of the set of fixed points,we see that ζ is isolated in E ∞ . So ( g, γ ) has no entry by (14.5), contradicting (i). Thisimplies that in fact s has no interior fixed points, so there is nothing to do. Case 3:
The last remaining alternative, namely when | α | < | β | , may be treated by simplyobserving that the fixed points for s are the same as the fixed points for s ∗ = ( β, g − , α ),and that s ∗ fits the previous case studied, so there are no interior fixed points for s ∗ , either.This concludes the proof that (i) and (ii) imply topological freeness. In order to provethat topological freeness implies (i), assume the former and suppose by contradiction thata G -circuit ( g, γ ) exists with no entry. Let x = r ( γ ), and notice that gx = gr ( γ ) = d ( γ ) , so the triple s := ( γ, g, x ) is seen to lie in S G,E . We may then use (14.2) to obtain a fixedpoint ζ for s , and by (14.5) we have that ζ is an isolated point of E ∞ , hence also an interiorfixed point.Working under the assumption of topological freeness, we deduce that ζ is a trivialfixed point, which is to say that there is an idempotent e in E , whose domain contains ζ , andsuch that se = e . Observing that e cannot possibly be zero, we deduce that e = ( ε, , ε ) , for some finite path ε . We then have that( ε, , ε ) = e = se = ( γ, g, x )( ε, , ε ) = (cid:0) γgε, ϕ ( g, ε ) , ε (cid:1) . In particular this implies that ε = γgε , so | ε | = | γgε | = | γ | + | gε | = | γ | + | ε | , whence | γ | = 0, contradicting the fact that G -circuits have nonzero length by definition.This shows that there are no G -circuit without an entry, hence proving (i).We next show that topological freeness implies (ii). So we suppose that some g in G pointwise fixes a whole cylinder Z ( x ), where x is a vertex. In particuler we have that gx = x , so the element s := ( x, g, x )belongs to S G,E , and it clearly also fixes every point in Z ( x ). Each ζ in Z ( x ) is thereforean interior fixed point for s , hence necessarily a trivial one by hypothesis. This meansthat there exists an idempotent e = ( γ, , γ ) ∈ E , such that ζ lies in the domain of e , alsoknown as Z ( γ ), and moreover se = e . Therefore( γ, , γ ) = e = se = ( x, g, x )( γ, , γ ) = (cid:0) xgγ, ϕ ( g, γ ) , γ (cid:1) , from where we deduce that gγ = γ , and ϕ ( g, γ ) = 1, which is to say that γ is stronglyfixed by g .Given that ζ ∈ Z ( γ ), we have that γ is a prefix of ζ , whence r ( γ ) = r ( ξ ) = x , so ζ ∈ Z ( γ ) ⊆ Z ( x ) . We then deduce that Z ( x ) is the union of the Z ( γ ), where γ range in the set of allfinite paths strongly fixed by g , with r ( γ ) = x . By compactness we may find a finitecollection of such finite paths, say γ , γ , . . . , γ k , such that Z ( x ) = k S i =1 Z ( γ i ) . (14 . . γ i ’s may be taken so that their length is constant.To see this let n be the length of the longer γ i , and observe that, for each i , one has that Z ( γ i ) = S r ( ε )= d ( γ i ) | ε | = n −| γ i | Z ( γ i ε ) . nverse semigroup actions and self-similar graphs 63 Moreover, each γ i ε occuring above is also strongly fixed by g , as seen in the discussionnear the beginning of section (5). Thus, if we replace each γ i by the set of all γ i ε , where ε is as above, all of the properties so far mentioned of the original γ i ’s will be preserved,and now | γ i ε | = | γ i | + | ε | = n. Therefore we may and will assume, from now on, that the γ i have a constant length,say n . From (14.10.2) it is now easy to conclude that the γ i exhaust the set of all finitepaths with range x and length n . In fact, if α is such a path, we may extend it to aninfinite path of the form ξ = αη . Since ξ ∈ Z ( x ), then ξ ∈ Z ( γ i ), for some i , whence γ i isa prefix of ξ and, by considering lengths, we see that α = γ i .The conclusion is that every finite path with length n and range x is strongly fixedby g , which is to say that g is slack at x . (cid:3) If for any g ∈ G \ { } and for any x ∈ E there exists η ∈ Z ( x ) suchthat gη = η , then (14.10.ii) holds trivially. This fact will be used in (18.9) and subsequentexamples. Regarding [ : Theorem 4.10.ii], and letting γ i be as in (14.10.2), onemay show that { f γ i } i is a cover of f x consisting of idempotents fixed under s (in the senseof [ : Definition 4.8.1]).In case ( G, E, ϕ ) is pseudo free, and if g is a nontrivial group element, then g admitsno strongly fixed paths by (5.5), so g will never be slack at any vertex. Condition (14.10.ii)can therefore only be satisfied if no nontrivial group element pointwise fixes a cylinder Z ( x ), and hence we have the following immediate consequence of (14.10): In addition to the conditions of (2.3), suppose that ( G, E, ϕ ) is pseudofree. Then the standard action of S G,E on E ∞ is topologically free if and only if thefollowing two conditions hold: (i) every G -circuit has an entry (which is the same as saying that every circuit has anentry by (14.6)), (ii) for every g in G , with g = 1 , and for every x in E , there is at least one ζ in Z ( x ) such that gζ = ζ . An important case for the theory of self-similar groups is when G acts faithfully on E ∞ , and E is a graph with a single vertex. Under the conditions of (2.3), suppose moreover that: (a) E has a single vertex, and at least two edges, (b) G acts faithfully on E ∞ .Then the standard action of S G,E on E ∞ is topologically free. Meaning that if gξ = ξ , for all ξ in E ∞ , then g = 1. Proof.
In the present situation the conditions of (14.10) become trivially true because: (i)all path are circuits and all circuits have entries, and (ii) there is only one Z ( x ) to consider,namely the whole space E ∞ , and by faithfulness no nontrivial group element acts triviallyon E ∞ . (cid:3) As the title of the present section suggests, our main interest is in determining condi-tions for G tight ( S G,E ) to be an essentially principal groupoid. Having understood topolog-ical freeness, an immediate consequence of [ : Theorem 4.7] is: Under the assumptions of (2.3), one has that G tight ( S G,E ) is essentiallyprincipal if and only if (14.10.i&ii) hold. Two other similar results could be stated giving conditions for G tight ( S G,E ) to beessentially principal, by combining [ : Theorem 4.7] with either (14.13) or (14.14), butwe will refrain from doing it here since the reader can easily guess them.
15. Local contractivity for S G,E . In [ : Section 6] local contractivity for groupoids and for actions of inverse semigroupsis studied. We will now use these results to characterize local contractivity for the tightgroupoid associated to an inverse semigroup S . Under the conditions of (2.3), one has that the following are equivalent: (i) S G,E is a locally contracting inverse semigroup, (ii) the standard action θ : S G,E y E ∞ is locally contracting, (iii) G tight ( S G,E ) is a locally contracting groupoid, (iv) every circuit in E has an entry. Proof.
As already mentioned in section (8), every tight filter in E is an ultra-filter, so theequivalence between (i) and (ii) follows from [ : Theorem 6.5].(ii) ⇒ (iii): Follows immediately from [ : Proposition 6.3].(iii) ⇒ (iv): We will prove this by contraposition, that is, assuming the existence of a circuit γ without an entry, we will show that G tight ( S G,E ) is not locally contracting.Our task is actually very easy. Given an entryless circuit γ , the path ξ = γγγ . . . isan isolated point, whence U := { ξ } is an open subset of E ∞ . Viewing the latter as theunit space of G tight ( S G,E ), as usual, and plugging U into [ : Definition 6.1], clearly therecan be no open set V , and bissection S , as mentioned there, simply because a chain ofnonempty subsets SV S − $ V ⊆ U cannot possibly exist withing a singleton such as U . This shows that G tight ( S G,E ) is notlocally contracting, as desired.(iv) ⇒ (i): Assuming that every circuit has an entry, we will show local contractivity of S G,E via [ : Proposition 6.7]. Given a nonzero idempotent e in E , write e = ( µ, , µ ) for somefinite path µ . Using that E has no sources, we may find an infinite path ξ = ξ ξ ξ . . . , nverse semigroup actions and self-similar graphs 65 such that d ( µ ) = r ( ξ ). Since E is a finite graph, there must be repetitions amongst the ξ i ,say ξ i = ξ j , for some i < j . Letting α = ξ ξ . . . ξ i , and γ = ξ i +1 ξ i +2 . . . ξ j , notice that d ( γ ) = d ( ξ j ) = d ( ξ i ) = r ( ξ i +1 ) = r ( γ ) , so γ is a circuit. It is also clear that µα and µαγ are well defined paths. Noticing that d ( α ) = r ( γ ) = d ( γ ) , we have that s := ( µαγ, , µα ) lies in S G,E . Moreover, setting β = µα, and using the notation introduced in (4.5), we have sf β s ∗ = ( µαγ, , µα )( µα, , µα )( µαγ, , µα ) ∗ (13.6.1) = ( µαγ, , µαγ ) ≤ f β , thus verifying [ : Proposition 6.7.ii]. By hypothesis γ has an entry, so we may find a path γ ′ , with r ( γ ′ ) = r ( γ ), which is not a prefix of γ , or vice versa. Setting β = µαγ ′ , we then have 0 = f µαγ ′ ≤ f µα ≤ f µ ⇒ = f β ≤ f β = s ∗ s ≤ e, verifying [ : Proposition 6.7.i]. Focusing now on [ : Proposition 6.7.iii] notice that f β s = ( µαγ ′ , , µαγ ′ )( µαγ, , µα ) = 0 , precisely because γ and γ ′ are not each other’s prefix. So evidently f β sf β = 0, provingthe last condition in [ : Proposition 6.7], and hence that S G,E is locally contracting, thusproving (i). (cid:3)
It is worth noticing that many results of [ ] used in the above proof, such as [ :Proposition 6.3], [ : Theorem 6.5] and [ : Proposition 6.7], comparing local contractivityfor groupoids, inverse semigroups, and actions, are either one way implications only, orthe converse depends on special conditions. Nevertheless, the situation in which we areworking has fortunately allowed for a downright equivalence of the various manifestationsof contractivity.However, this result should be taken with a certain skepticism. First of all it is wellknown that the above condition on circuits is not sufficient for local contractivity for thegroupoid associated to infinite graphs [ ]. Considering that finite graphs are special casesof our theory (just take the acting group to be the trivial group), it is not unreasonable to believe that our results admit natural generalizations to infinite graphs, but then acharacterization of local contractivity for the corresponding groupoid will certainly notfollow from the fact that every circuit has an entry, since this is false for infinite graphs,as mentioned above.Secondly, observe that the condition on the existence of entries for circuits completelyignores the group G , but, again, a generalization to infinite graphs will probably dependon the action. A hypothesis such as “every vertex connects to a G -circuit with an entry”,to paraphrase the main hypothesis of [ : Lemma 3.8], is probably more realistic in theconjectured infinite graph scenario.
16. Simplicity and pure infiniteness for O G,E . In this section we use the results in the previous sections to characterize when O G,E issimple and purely infinite. The central results are the following:
Assume that ( G, E, ϕ ) satisfies (2.3), that G is amenable, and that forevery g ∈ G there are at most finitely many minimal strongly fixed paths for g . Then O G,E is simple if and only if the following conditions are satisfied: (a) E is weakly- G -transitive. (b) Every G -circuit has an entry. ( c ) Given a vertex x , and a group element g fixing Z ( x ) pointwise, then necessarily g isa slack at x . Proof.
By (12.2), the groupoid G tight ( S G,E ) is Hausdorff. Clearly G tight ( S G,E ) is ´etalewith second countable unit space. By (10.18), G tight ( S G,E ) is amenable. Then, by (6.4), O G,E ∼ = C ∗ ( G tight ( S G,E )) = C ∗ r ( G tight ( S G,E )). By [ : Theorem 5.1], O G,E is simple if andonly if G tight ( S G,E ) is minimal and essentially principal. Since minimality of G tight ( S G,E ) isequivalent to (a) by (13.6), and essential principality of G tight ( S G,E ) is equivalent to (b&c)by (14.15), the result holds. (cid:3)
With respect to pure infiniteness, we have:
Let ( G, E, ϕ ) be under (2.3), and let G be an amenable group. If G tight ( S G,E ) is essentially principal, then every hereditary subalgebra of O G,E contains aninfinite projection.
Proof.
By the same argument as in (16.1), O G,E ∼ = C ∗ ( G tight ( S G,E )) = C ∗ r ( G tight ( S G,E )).By (14.10.i), every circuit of E has an entry. Thus, G tight ( S G,E ) is locally contracting by(15.1). Hence, by [ : Proposition 2.4], every nonzero hereditary sub-C*-algebra of O G,E contains an infinite projection, as desired. (cid:3)
As an immediate consequence we have If ( G, E, ϕ ) satisfies (2.3), the group G is amenable, and G tight ( S G,E ) is Hausdorff then, whenever O G,E is simple, it is necessarily also purely infinite (simple).
Proof.
By (14.10) and (16.1), G tight ( S G,E ) is essentially principal. Thus, by (16.2), everynonzero hereditary sub-C*-algebra of O G,E contains an infinite projection. Hence, O G,E is purely infinite simple, as desired. (cid:3) nverse semigroup actions and self-similar graphs 67
17. Revisiting Nekrashevych algebras.
In this section we will analyze Nekrashevych algebras from our point of view.The Nekrashevych C*-algebra O ( G,X ) , associated to a self-similar action of a group G on a finite alphabet X [ ], is a direct example of our definition (see (3.3)). Here, thegraph E is the rose of n petals for n = | X | ≥
2, so that the action on vertices is trivial,and the action is faithful. Since | E | = 1, we have the following facts:(1) E is G -transitive, whence G tight ( S G,E ) is minimal by (13.6).(2) G tight ( S G,E ) is essentially principal by (14.14) and [ : Theorem 14.7]. In particular, G tight ( S G,E ) is locally contracting by (14.10) and (15.1).Thus, if G tight ( S G,E ) is Hausdorff, we conclude that C ∗ r ( G tight ( S G,E )) is a purely infinitesimple C*-algebra by [ : Theorem 6.8]. Hausdorffness of G tight ( S G,E ) is equivalent, ac-cording to (12.2), to the existence of at most finitely many minimal strongly fixed pathsfor every g ∈ G .In this sense, it is interesting to remark that Nekrashevych also gave a presentationof its algebra as a groupoid C*-algebra associated to a groupoid of germs of an inversesemigroup S [ : Section 5]. While S turns out to be S G,E , the notion of germ that heused is the one adopted by Arzumanian and Renault [ ], which differs from the one weused, due to Patterson [ : Page 140]. Luckily, both definitions coincide when the action of S G,E on E ∞ is topologically free, which is the case of Nekrashevych triples, as we noticedabove. So, Nekrashevych’s groupoid and G tight ( S G,E ) coincide, and the characterization ofHausdorffness we obtained in (12.2) coincide with that given by Nekrashevych [ : Lemma5.4]. In order to obtain a characterization of (pure infinite) simplicity for O ( G,X ) , we needto keep control of whether O ( G,X ) is nuclear. So, it only remains to determine when G tight ( S G,E ) is amenable, which implies that C ∗ ( G tight ( S G,E )) = C ∗ r ( G tight ( S G,E )). By(10.18), if G is an amenable group, then G tight ( S G,E ) is an amenable groupoid. Thus, weobtain If ( G, X, ϕ ) is a Nekrashevych triple, with G an amenable groupand G tight ( S G,E ) a Hausdorff groupoid, then O ( G,X ) is a nuclear, separable, purely infinitesimple C*-algebra. Here, Nekrashevych’s approach differs from ours. In [ ] he stated a sufficient con-dition for the amenability of G tight ( S G,E ), which apparently does not require the group G to be amenable. The condition relies on two concepts associated to self-similar groups:self-replication and contractiveness (see [ ] or [ ] for definitions of these concepts).Nekrashevych [ : Theorem 5.6] proved that if G tight ( S G,E ) is Hausdorff and (
G, X ) isself-replicating and contractive, then G tight ( S G,E ) is of polynomial growth [ ], and thusit its amenable by [ : Proposition 3.2.32].
18. Revisiting Katsura algebras.
In this section we will analyze Katsura algebras from our point of view.We will quickly recall the definition and basic properties of Katsura algebras that willbe needed in the sequel. This is borrowed from [ ]. Let N ∈ N ∪ {∞} , let A ∈ M N ( Z + ) and B ∈ M N ( Z ) be row-finitematrices. Define a set Ω A byΩ A := { ( i, j ) ∈ { , , . . . , N } × { , , . . . , N } : A i,j ≥ } . For each i ∈ { , , . . . , N } , define a set Ω A ( i ) ⊂ { , , . . . , N } byΩ A ( i ) := { j ∈ { , , . . . , N } : ( i, j ) ∈ Ω A } . Notice that, by definition, Ω A ( i ) is finite for all i . Finally, fix the following relation:(0) Ω A ( i ) = ∅ for all i , and B i,j = 0 for ( i, j ) Ω A .With these data we can define Katsura algebras Define O A,B to be the universal C*-algebra generated by mutuallyorthogonal projections { q i } Ni =1 , partial unitaries { u i } Ni =1 with u i u ∗ i = u ∗ i u i = q i , andpartial isometries { s i,j,n } ( i,j ) ∈ Ω A ,n ∈ Z satisfying the relations:(i) s i,j,n u j = s i,j,n + A i,j and u i s i,j,n = s i,j,n + B i,j for all ( i, j ) ∈ Ω A and n ∈ Z .(ii) s ∗ i,j,n s i,j,n = q j for all ( i, j ) ∈ Ω A and n ∈ Z .(iii) q i = P j ∈ Ω A ( i ) A i,j P n =1 s i,j,n s ∗ i,j,n for all i . Now, the following facts holds:(1) The C*-algebra O A,B is separable, nuclear, in the UCT class [ : Proposition 2.9].(2) If the matrices A, B satisfy the following additional properties:(a) A is irreducible, and(b) | A ii | ≥ B i,i = 1 for every 1 ≤ i ≤ N ,then the C*-algebra O A,B is purely infinite simple, and hence a Kirchberg algebra[ : Proposition 2.10].(3) The K -groups of O A,B are [ : Proposition 2.6]:(a) K ( O A,B ) ∼ = coker( I − A ) ⊕ ker( I − B ), and(b) K ( O A,B ) ∼ = coker( I − B ) ⊕ ker( I − A ).(4) Every Kirchberg algebra can be represented, up to isomorphism, by an algebra O A,B for matrices
A, B satisfying the conditions in (18.3.2) [ : Proposition 4.5].As we have seen in (3.4), unital Katsura algebras are natural examples of our con-struction. So, it is easy to use our results in order to characterize some properties, likesimplicity or pure infinite simplicity, in terms of matrices A and B . This work has beenpreviously done in [ ], but the approach we chose there was fairly more direct and com-putational, so that the conditions appearing there were less elegant and clear than theones we will present here.Across this section, we will say that a triple ( Z , E, ϕ ) is a Katsura triple if there existfinite matrices A, B satisfying (18.1) such that the triple associated to the algebra O A,B is nverse semigroup actions and self-similar graphs 69 ( Z , E, ϕ ); in particular, E is the graph whose adjacency matrix is A . Also, we will fix thefollowing agreement: let ξ be either in E ∗ or in E ∞ , i.e. ξ = e i ,i ,n e i ,i ,n · · · e i k ,i k +1 ,n k or ξ = e i ,i ,n e i ,i ,n · · · e i k ,i k +1 ,n k · · · , then, for any r ∈ N we define B ξ | r := r Y t =1 B i t ,i t +1 and A ξ | r := r Y t =1 A i t ,i t +1 . The first step to work out the corresponding results to the ones we obtained for thegeneral setting is to determine when a finite path α ∈ E ∗ is fixed under the action of anelement l ∈ Z . Let ( Z , E, ϕ ) be a Katsura triple. Given an element α of E ∗ of length r and an integer l ∈ Z , the following are equivalent: (1) α is fixed under the action of l . (2) For every ≤ j ≤ r the element K j := l B α | j A α | j belongs to Z . Proof.
Set α = e i ,i ,n e i ,i ,n · · · e i r ,i r +1 ,n r . By definition of ( Z , E, ϕ ), α = lα if and onlyif there exists a sequence ( K j ) j ≥ ⊆ Z such that:(i) K = l .(ii) For every 1 ≤ j ≤ r , n j − + K j − B i j ,i j +1 = n j − + K j A i j ,i j +1 .Notice that (ii) is equivalent to ask K j − B i j ,i j +1 = K j A i j ,i j +1 for every j ≥ j = 1 we have K B i ,i = lB i ,i = K A i ,i , so that K = l B i ,i A i ,i . Now,suppose that for 1 ≤ t ≤ j − K t := l B α | t A α | t . Hence K j A i j ,i j +1 = K j − B i j ,i j +1 = l B α | j − A α | j − · B i j ,i j +1 , so that K j = l B α | j A α | j . This completes the proof. (cid:3) Now, we are ready to characterize pseudo freeness for a Katsura triple ( Z , E, ϕ ). Let ( Z , E, ϕ ) be a Katsura triple. Then, the following are equivalent: (1) ( Z , E, ϕ ) is pseudo free. (2) B i,j = 0 if and only if ( i, j ) Ω A . Proof.
Let α = e i ,i ,n e i ,i ,n · · · e i r ,i r +1 ,n r of E ∗ , and let l ∈ Z . By (18.4), lα = α exactlywhen the elements K j := l B α | j A α | j belongs to Z for every 1 ≤ j ≤ r . Since ϕ ( l, α | j ) = K j , ϕ ( l, α ) = 0 exactly when K j = 0 for some j ≤ r . Thus, the situation reduces to K j − e i j ,i j +1 ,n l = e i j ,i j +1 ,n l and K j = 0 for some 1 ≤ j ≤ r , which corresponds to the equa-tion n j + K j − B i j ,i j +1 = n j . And this occurs exactly when B i j ,i j +1 = 0, so we are done. (cid:3) Which these results in mind, we are ready to characterize when G tight ( S Z ,E ) is Haus-dorff Let ( Z , E, ϕ ) be a Katsura triple. Then, the following are equivalent: (1) G tight ( S Z ,E ) is Hausdorff. (2) Whenever ( i, j ) ∈ Ω A with B i,j = 0 , then for any l ∈ Z there exist finitely many finitepaths α ∈ E ∗ with d ( α ) = i such that l B α | t A α | t ∈ Z for every ≤ t ≤ r − . Proof.
The result holds by (12.2) and (18.5). (cid:3)
The next step is to determine the minimality of G tight ( S Z ,E ). Let ( Z , E, ϕ ) be a Katsura triple. Then, the following are equivalent: (1) G tight ( S Z ,E ) is minimal. (2) The adjacency matrix A of E is irreducible. Proof.
First notice that E has no sinks by (18.1.(0)). Moreover, the action of Z on E fixesall the vertices. Then, by (13.6), G tight ( S Z ,E ) is minimal if and only if E is transitive,which is equivalent to the matrix A being irreducible, so we are done. (cid:3) Now, we will give a characterization of when G tight ( S Z ,E ) is essentially principal. Let ( Z , E, ϕ ) be a Katsura triple. Then, the following are equivalent: (1) G tight ( S Z ,E ) is essentially principal. (2) (a) Every circuit in E has an entry. (b) If ≤ i ≤ N , l ∈ Z , and for any ξ ∈ Z ( i ) the elements l B ξ | n A ξ | n ∈ Z for all n ∈ N , then there exists m ∈ N such that B ξ | m = 0 for all ξ ∈ Z ( i ) . Proof.
Since the action of Z fixes all the vertices of E , (2a) is (14.10.i). On the other side,(2b) is exactly (14.10.ii) because of (18.4) and (18.5). Thus, the result is consequence of(14.15). (cid:3) We can obtain an easy sufficient condition for G tight ( S Z ,E ) being essentially principal. Let ( Z , E, ϕ ) be a Katsura triple. If (1) Every circuit of E has an entry, and (2) For every ≤ i ≤ N and every l ∈ Z there exists η ∈ Z ( i ) such that lim n →∞ l B η | n A η | n = 0 ,then G tight ( S Z ,E ) is essentially principal. nverse semigroup actions and self-similar graphs 71 Proof.
By (18.4), condition (2) implies that lη = η for any l ∈ Z , whence the triple( Z, E, ϕ ) trivially satisfies (14.10.ii), as remarked in (14.11). (cid:3)
Corollary (18.9) applies when we have a pair of finite matrices
A, B under (18.1), suchthat for every 1 ≤ i ≤ N we have | A ii | ≥ B ii < | A ii | . In particular, G tight ( S Z ,E ) isessentialy principal for Katsura systems ( Z , E, ϕ ) satisfying (18.3.2).Also, it is immediate to characterize when G tight ( S Z ,E ) is locally contracting. Let ( Z , E, ϕ ) be a Katsura triple. Then, the following are equivalent: (1) G tight ( S Z ,E ) is locally contracting. (2) Every circuit of E has an entry. Proof.
This is (15.1). (cid:3)
Finally, we have the following fact If ( Z , E, ϕ ) is a Katsura triple, then G tight ( S Z ,E ) is an amenablegroupoid. Proof.
Since Z is an amenable group, (10.18) applies. (cid:3) Now, we are ready to characterize simplicity of the algebra O A,B , as follows
Let ( Z , E, ϕ ) be a Katsura triple such that G tight ( S Z ,E ) is Hausdorff(see (18.6)). Then, the following are equivalent: (1) (a) The matrix A is irreducible. (b) Every circuit of E has an entry. ( c ) If ≤ i ≤ N , l ∈ Z , and for any ξ ∈ Z ( i ) the elements l B ξ | n A ξ | n ∈ Z for all n ∈ N , then there exists m ∈ N such that B ξ | m = 0 . (2) O A,B is simple.
Proof.
This is exactly (16.1) for the Katsura triple ( Z , E, ϕ ), because of (18.7), (18.8) and(18.11). (cid:3) In particular, when G tight ( S Z ,E ) is Hausdorff and O A,B is simple, the G tight ( S Z ,E ) islocally contracting by (18.10) and (18.12.1b). Hence, we have If ( Z , E, ϕ ) is a Katsura triple such that G tight ( S Z ,E ) is Hausdorff and O A,B is simple, then O A,B is purely infinite simple.
Proof.
This is by (16.3). (cid:3)
Notice that, because of (18.9), Katsura’s condition (18.3.2) for O A,B being a purely infinite simple C*-algebra derive directly from (18.12) and (18.13) when G tight ( S Z ,E ) is Hausdorff. Moreover, when G tight ( S Z ,E ) is Hausdorff, (18.12) provides acharacterization of simplicity for O A,B , improving Katsura’s results on that direction,where only sufficient conditions are given [ ]. We close this section by presenting a couple of examples. The first one illustrates thedifference between (
G, E, ϕ ) being pseudo free and G tight ( S G,E ) being Hausdorff, and alsothe difference between the action of S G,E on E ∞ being topologically free and the actionof G on E ∞ being topologically free. Set N = 2, and consider the matrices A = (cid:18) (cid:19) and B = (cid:18) (cid:19) .Let ( Z , E, ϕ ) be the associated Katsura triple. Then, we have the following:(1) Since Z is amenable, then so is G tight ( S Z ,E ).(2) Since A is irreducible, G tight ( S Z ,E ) is minimal.(3) Every circuit in E has an entry.(4) Since A , = 0 and B , = 0, ( Z , E, ϕ ) is not pseudo free by (18.5).(5) Notice that the only possible quotient values B i,j A i,j are B , A , = B , A , = 12 and B , A , = B , A , = 01 = 0. Then, for any l ∈ Z , it is clear that there exists only finitely manyminimal strongly fixed paths for l . Thus, G tight ( S Z ,E ) is Hausdorff by (18.6).(6) Moreover, by the argument in point (5), the only infinite paths fixed by the action of Z are the ones associated to minimal strongly fixed paths, and thus trivial. Hence,the action of S Z ,E on E ∞ is topologically free. But the action of Z is not topologicallyfree, since every element of Z fix the cylinders Z ( e , , ) and Z ( e , , ).Notice that O A,B is purely infinite simple by (18.12) and (18.13).The second example shows that (
G, E, ϕ ) being pseudo free do not imply that theaction of S G,E is topologically free.
Set N = 1, and set A = B = ( n ) for any n ≥
2. Let ( Z , E, ϕ ) be theassociated Katsura triple. Then, we have the following:(1) Since Z is amenable, then so is G tight ( S Z ,E ).(2) Since A is irreducible, G tight ( S Z ,E ) is minimal.(3) Every circuit in E has an entry.(4) Since A = B , ( Z , E, ϕ ) is pseudo free, whence in particular G tight ( S Z ,E ) is Hausdorff.(5) Since A = B , the action of Z on E is trivial. Thus, the action of S Z ,E on E ∞ (and of Z ) cannot be topologically free, because there are no slacks. References [1] C. Anantharaman-Delaroche, “Purely infinite C ∗ -algebras arising form dynamical systems”, Bull.Soc. Math. France, (1997), no. 2, 199–225.[2] C. Anantharaman-Delaroche and J. Renault, “Amenable groupoids”, Monogr. Enseign. Math. 36,Universit´e de Gen`eve, 2000.[3] V. Arzumanian, J. Renault, “Examples of pseudogroups and their C ∗ -algebras”, Operator algebrasand quantum field theory (Rome, 1996), 93-104, Int. Press, Cambridge, MA, 1997.[4] J. Brown, L. O. Clark, C. Farthing and A. Sims, “Simplicity of algebras associated to ´etale groupoids”, Semigroup Forum, (2014), 433–452. nverse semigroup actions and self-similar graphs 73 [5] N. P. Brown and N. Ozawa, “C*-algebras and finite-dimensional approximations”, Graduate Studiesin Mathematics, 88, American Mathematical Society, 2008.[6] R. Exel, “Inverse semigroups and combinatorial C*-algebras”, Bull. Braz. Math. Soc., (2008), no.2, 191–313.[7] R. Exel, “Non-Hausdorff ´etale groupoids”, Proc. Amer. Math. Soc., (2011), no. 3, 897–907.[8] R. Exel, “Partial Dynamical Systems, Fell Bundles and Applications”, Licensed under a CreativeCommons Attribution-ShareAlike 4.0 International License, 351pp, 2014. Available online frommtm.ufsc.br/ ∼ exel/publications. PDF file md5sum: bc4cbce3debdb584ca226176b9b76924.[9] R. Exel and M. Laca, “Cuntz-Krieger algebras for infinite matrices”, J. reine angew. Math., (1999), 119–172.[10] R. Exel and E. Pardo, “Representing Kirchberg algebras as inverse semigroup crossed products”,arXiv:1303.6268 [math.OA], 2013.[11] R. Exel and E. Pardo, “Graphs, groups and self-similarity”, arXiv:1307.1120 [math.OA], 2013.[12] R. Exel and E. Pardo, “The tight groupoid of an inverse semigroup”, arXiv:1408.5278 [math.OA],2014.[13] R. Exel and C. Starling, “Self-similar graph C*-algebras and partial crossed products”, arXiv:1406.1086 [math.OA], 2014.[14] R. Exel and A. Vershik, “C*-algebras of irreversible dynamical systems”,
Canadian MathematicalJournal, (2006), 39–63.[15] R. I. Grigorchuk, “On Burnside’s problem on periodic groups”, Funct. Anal. Appl., (1980), 41–43.[16] N. D. Gupta and S. N. Sidki, “On the Burnside problem for periodic groups”, Math. Z., (1983),385–388.[17] T. Katsura, “A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras. I. Fundamental results”,
Trans. Amer. Math. Soc., (2004), no. 11, 4287–4322.[18] T. Katsura, “A construction of actions on Kirchberg algebras which induce given actions on their K -groups”, J. reine angew. Math., (2008), 27–65.[19] T. Katsura, “A class of C ∗ -algebras generalizing both graph algebras and homeomorphism C ∗ -algebras IV, pure infiniteness”, J. Funct. Anal., (2008), 1161–1187.[20] A. Kumjian, D. Pask, I. Raeburn and J. Renault, “Graphs, groupoids, and Cuntz-Krieger algebras”,
J. Funct. Anal., (1997), 505–541.[21] A. Kumjian, D. Pask and I. Raeburn, “Cuntz-Krieger algebras of directed graphs”,
Pacific J. Math., (1998), no. 1, 161–174.[22] M. V. Lawson, “Inverse semigroups, the theory of partial symmetries”, World Scientific, 1998.[23] M. V. Lawson, “Compactable semilattices”,
Semigroup Forum, (2010), no. 1, 187–199.[24] V. Nekrashevych, “Cuntz-Pimsner algebras of group actions”, J. Operator Theory, (2004), 223–249.[25] V. Nekrashevych, “Self-similar groups”, Mathematical Surveys and Monographs, Amer. Math.Soc., Providence, RI, 2005.[26] V. Nekrashevych, “C*-algebras and self-similar groups”,
J. reine angew. Math., (2009), 59–123.[27] A. L. T. Paterson, “Groupoids, inverse semigroups, and their operator algebras”, Birkh¨auser, 1999.[28] G. K. Pedersen, “C*-algebras and Their Automorphism Groups”, Academic Press, 1979.[29] M. V. Pimsner, “A class of C*-algebras generalizing both Cuntz-Krieger algebras and crossed productsby Z ”, Fields Inst. Commun., (1997), 189–212.[30] I. Raeburn, “Graph algebras”, CBMS Regional Conference Series in Mathematics, (2005), pp.vi+113.[31] J. Renault, “Cartan subalgebras in C ∗ -algebras”, Irish Math. Soc. Bull., (2008), 29–63.[32] C. Starling, “Boundary quotients of C*-algebras of right LCM semigroups”, arXiv:1409.1549[math.OA]. [33] B. Steinberg, “A groupoid approach to discrete inverse semigroup algebras”, Adv. Math., (2010),689–727.
Departamento de Matem´atica; Universidade Federal de Santa Catarina;88010-970 Florian´opolis SC; Brazil ([email protected])