aa r X i v : . [ m a t h . OA ] M a y A DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS HOSSEIN LARKI
Abstract.
We investigate the pure infiniteness and stable finitenessof the Exel-Pardo C ∗ -algebras O G,E for countable self-similar graphs(
G, E, ϕ ). In particular, we associate a specific ordinary graph e E to( G, E, ϕ ) such that some properties such as simpleness, stable finitenessor pure infiniteness of the graph C ∗ -algebra C ∗ ( e E ) imply that of O G,E .Among others, this follows a dichotomy for simple O G,E : if (
G, E, ϕ )contains no G -circuits, then O G,E is stably finite; otherwise, O G,E ispurely infinite.Furthermore, Li and Yang recently introduced self-similar k -graph C ∗ -algebras O G, Λ . We also show that when | Λ | < ∞ and O G, Λ issimple, then it is purely infinite. Introduction
In [7], Exel and Pardo introduced self-similar graph C ∗ -algebras O G,E togive a unified framework like graph C ∗ -algebras for the Katsura’s [10] andNekrashevych’s algebras [18, 19]. These C ∗ -algebras were initially consid-ered in [7] only for countable discrete groups G acting on finite graphs E with no sources, and then generalized in [2, 8] for larger classes. Roughlyspeaking, Exel and Pardo attached an inverse semigroup S G,E and thetight groupoid G tight ( S G,E ) to (
G, E, ϕ ) such that O G,E ∼ = C ∗ ( G tight ( S G,E )),and then describe amenability [7, Corollary 10.18], minimality [7, Theo-rem 13.6], and effectivity (or topological principality) [7, Corollary 14.15] of G tight ( S G,E ), and thus simplicity and pure infiniteness of O G,E [7, Section16], among others. Although only finite graphs are considered in [7], butmany arguments and proofs work for countable row-finite graphs with nosources (see [8]).The initial aim of this note comes from a dichotomy for simple groupoid C ∗ -algebras [21, 3]. According to [21, Theorem 4.7] and [3, Corollary 5.13],a simple reduced C ∗ -algebra C ∗ r ( G ) of ample groupoid G with an almostunperforated type semigroup is either purely infinite or stable finite. Weexplicitly describe this dichotomy for self-similar graph C ∗ -algebras O G,E
Date : May 13, 2020.2010
Mathematics Subject Classification.
Key words and phrases. self-similar graph, C ∗ -algebra, pure infiniteness, stablefiniteness. by the underlying graphical properties. Here, we consider countable row-finite source-free graphs E over an amenable (countable) group G [2, 8].However, our results may be generalized to any countable graph E by thedesingularization of [8].We begin in Section 2 by reviewing necessary background on groupoid andself-similar graph C ∗ -algebras. Then, in Section 3, we generalize the Exel-Pardo’s characterization of purely infinite O G,E to countable self-similargraphs by the groupoid approach (for not necessarily simple cases). More-over, for certain self-similar graphs (
G, E, ϕ ), we show that the C ∗ -algebra O G,E is purely infinite and simple if and only if the additive monoid ofnonzero Murray-von Neumann equivalent projections in M ∞ ( O G,E ) is agroup.In Section 4, we focus on the stable finiteness of O G,E . We attach a spacialgraph e E to ( G, E, ϕ ) such that some properties of O G,E - such as simplicity,pure infiniteness, and stable infiniteness- can be derived from those of thegraph C ∗ -algebra C ∗ ( e E ). Then using known results about the graph C ∗ -algebras, we show that a simple C ∗ -algebra O G,E is stable finite if and onlyif the underlying (
G, E, ϕ ) contains no G -circuits. In particular, we deducea dichotomy: A simple O G,E is purely infinite if (
G, E, ϕ ) has a G -circuit;otherwise, it is stable finite.As the k -graph version of Exel-Pardo C ∗ -algebras, Li and Yang introducedself-similar k -graphs ( G, Λ) and associated C ∗ -algebras O G, Λ . Briefly, by agroupoid approach, they investigated their properties such as nuclearity [17,Theorem 6.6(i)], amenability [17, Theorem 5.9], and simplicity [17, Theorem6.6(ii)]. In Section 5, We investigate the pure infiniteness of O G, Λ for thenonsimple cases. In particular, we modify and extend [17, Theorem 6.13]. Acknowledgement.
The author appreciates Enrique Pardo for review-ing the initial version of the article and his helpful comments; in particular,for noting a gap in the proof of Theorem 3.8.2.
Preliminaries
Groupoid C ∗ -algebras. We give here a brief introduction to amplegroupoids and associated C ∗ -algebras; for more details see [23, 1] for exam-ple. A groupoid is a small category G with inverses. The unit space of G is the set of identity morphisms, that is G (0) := { α − α : α ∈ G} . For each α ∈ G , we may define the range r ( α ) := αα − and the source s ( α ) := α − α ,which satisfy r ( α ) α = α = αs ( α ). Hence, for α, β ∈ G , the composition αβ is well-defined in G if and only if s ( α ) = r ( β ). The isotropy subgroupoid of G is defined by Iso( G ) := { α ∈ G : s ( α ) = r ( α ) } . We work usually with groupoids G endowed with a topology such thatthe maps r, s : G → G (0) are continuous (in this case, G is called a topologicalgroupoid ). A subset B ⊆ G is called a bisection if both restrictions r | B and DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 3 s | B are homeomorphisms. We say that G is ample in case G has a basis ofcompact and open bisections. Definition . Let G be a topological groupoid. We say that G is effective ifthe interior of Iso( G ) is just G (0) . Moreover, G is called topologically principal if { u ∈ G (0) : s − ( u ) ∩ r − ( u ) = { u }} is dense in G (0) .Note that, when G is second-countable, [22, Proposition 3.3] implies that G is effective if and only if it is topologically principal. In this paper, wewill work frequently with second-countable effective ample groupoids.We now recall the definition of reduced C ∗ -algebra C ∗ r ( G ). Let G be anample groupoid. We write C c ( G ) for the complex vector space consisting ofcompactly supported continuous functions on G , which is an ∗ -algebra withthe convolution multiplication and the involution f ∗ ( α ) := f ( α − ). For eachunit u ∈ G (0) and G u := s − ( { u } ), let π u : C c ( G ) → B ( ℓ ( G u )) be the leftregular ∗ -representation defined by π u ( f ) δ α := X s ( β )= r ( α ) f ( β ) δ βα ( f ∈ C c ( G ) , α ∈ G u ) . Then the reduced C ∗ -algebra C ∗ r ( G ) is the completion of C c ( G ) under thereduced C ∗ -norm k f k r := sup u ∈G (0) k π u ( f ) k . Moreover, there is a full C ∗ -algebra C ∗ ( G ) associated to G , which is thecompletion of C c ( G ) taken over all k . k C c ( G ) -decreasing representations of G .Hence, C ∗ r ( G ) is a quotient of C ∗ ( G ), and [1, Proposition 6.1.8] shows thatthey are equal if the underlying groupoid G is amenable. Definition . We say that a C ∗ -algebra A is purely infinite if everynonzero hereditary C ∗ -subalgebra of A contains an infinite projection.The following is analogous to [4, Theorem 4.1] without the minimalityassumption. Proposition 2.3.
Let G be a second-countable Hausdorff ample groupoidand let B be a basis of compact open sets for G (0) . Suppose also that G iseffective. Then C ∗ r ( G ) is purely infinite if and only if V is infinite in C ∗ r ( G ) for every V ∈ B ( V is the characteristic function of V ).Proof. The “only if” implication is immediate. For the converse, supposethat every 1 V in C ∗ r ( G ) is infinite for V ∈ B . Let A be a nonzero hereditary C ∗ -subalgebra of C ∗ r ( G ) and take some positive element 0 = a ∈ A . Usingthe hereditary property, we may follow the proof of [15, Proposition 5.2] tofind a projection p ∈ A and some V ∈ B such that p ∼ V in the Murray-vonNuemann sense. Since the infiniteness is preserved under ∼ , then p is aninfinite projection, concluding the result. (cid:3) HOSSEIN LARKI
Graph C ∗ -algebras. Let E = ( E , E , r, d ) be a directed graph withthe vertex set E , the edge set E , and the range and domain maps r, d : E → E . We say that E is row-finite if each vertex receives at most finitelymany edges. A source in E is a vertex v ∈ E which receives no edges, i.e. d − ( v ) = ∅ . We will write by E ∗ the set of finite paths in E , that is E ∗ := [ n ≥ E n = [ n ≥ { α = e . . . e n : e i ∈ E , d ( e i ) = r ( e i +1 ) } . Then one may extend r, d : E ∗ → E by defining r ( α ) = r ( e ) and d ( α ) = d ( e n ) for every path α = e . . . e n ∈ E n . Throughout the paper, we willconsider only countable directed graphs.Given a directed graph E , a Cuntz-Krieger E -family is a collection { p v , s e : v ∈ E , e ∈ E } of pairwise orthogonal projections p v and partial isometries s e with the following relations(1) s ∗ e s e = p d ( e ) for every e ∈ E ,(2) s e s ∗ e ≤ p r ( e ) for every e ∈ E , and(3) p v = P d ( e )= v s e s ∗ e for all vertices v with 0 < | d − ( v ) | < ∞ .The graph C ∗ -algebra C ∗ ( E ) is the universal C ∗ -algebra generated by aCuntz-Krieger E -family { p v , s e } [20]. By the above relations, for e , . . . , e n ∈ E , s e . . . s e n is nonzero if and only if α := e . . . e n is a path in E ; in thiscase, we write s α := s e . . . s e n .2.3. Self-similar graphs and their C ∗ -algebras. Let G be a countablediscrete group. An action G y E is a map G × ( E ∪ E ) → E ∪ E ,denoted by ( g, a ) ga , such that the action of each g ∈ G on E gives agraph automorphism.A self-similar graph is a triple ( G, E, ϕ ) such that(1) E is a directed graph,(2) G acts on E by automorphisms, and(3) ϕ : G × E → G is a 1-cocycle for G y E satisfying ϕ ( g, e ) v = gv for every g ∈ G , e ∈ E , and v ∈ E . Remark . According to [7, Proposition 2.4], we may extend inductivelythe action G y E and the cocycle ϕ on the finite path space E ∗ satisfyingthe desired relations [7, Equation 2.6]. Indeed, if α = α α ∈ E ∗ , then wedefine gα = ( gα )( ϕ ( g, α ) α ) and ϕ ( g, α ) = ϕ ( ϕ ( g, α ) , α ) . Definition . Let (
G, E, ϕ ) be a (countable) self-similar graph.Then O G,E is the universal C ∗ -algebra generated by { p v , s e : v ∈ E , e ∈ E } ∪ { u g p v : g ∈ G, v ∈ E } satisfying the following properties:(1) { p v , s e : v ∈ E , e ∈ E } is a Cuntz-Krieger E -family. DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 5 (2) u : G → M ( O G,E ), g u g , is a unitary ∗ -representation of G onthe multiplier algebra M ( O G,E ).(3) u g p v = p gv u g for every g ∈ G and v ∈ E .(4) u g s e = s ge u ϕ ( g,e ) for every g ∈ G and e ∈ E .We usually use the notation O G,E instead of O ( G,E,ϕ ) for convenience. Also,we will write each u g p v by u gv . Then one may easily verify relations (b)-(e)of [8, Definition 2.2]. Standing assumption.
All self-similar graphs (
G, E, ϕ ) considered inthis paper will be countable, row-finite and source-free.2.4.
The groupoid associated to ( G, E, ϕ ) . In [7, Section 4], Exel andPardo associated an inverse semigroup S G,E to a self-similar graph (
G, E, ϕ )with finite graph E . They then showed that O G,E ∼ = C ∗ tight ( S G,E ) ∼ = C ∗ ( G G,E )where G G,E is the groupoid of germs for the action of S G,E on E ∞ [7, Corol-lary 6.4 and Proposition 8.4]. Note that the constructions of S G,E and G G,E in [7] may be extended for countable row-finite, source-free self-similargraphs (
G, E, ϕ ) with small modifications. We give a brief review of it herefor convenience. So, fix a row-finite self-similar graph (
G, E, ϕ ) withoutsources. Define the ∗ -inverse semigroup S G,E as S G,E = { ( α, g, β ) : α, β ∈ E ∗ , g ∈ G, d ( α ) = gd ( β ) } ∪ { } with the operations( α, g, β )( γ, h, δ ) := ( α, gϕ ( h, ε ) , δhε ) if β = γε ( αgε, ϕ ( g, ε ) h, δ ) if γ = βε α, g, β ) ∗ := ( β, g − , α ).Let E ∞ be the space one-sided infinite paths of the form x = e e . . . such that d ( e i ) = r ( e i +1 ) for i ≥ . By [7, Proposition 8.1], there is a unique action G y E ∞ as follows: for each g ∈ G and x = e e . . . ∈ E ∞ , there is a unique infinite path gx = f f . . . such that f f . . . f n = g ( e e . . . e n ) (for all n ≥ . Moreover, we may consider the action of each ( α, g, β ) ∈ S G,E on x = β ˆ x ∈ E ∞ by ( α, g, β ) · x = α ( g ˆ x ). Then G G,E is the groupoid of germs of theaction of S G,E on E ∞ , that is G G,E = (cid:8) [ α, g, β ; x ] : x = β ˆ x (cid:9) . Recall that two germs [ s ; x ] , [ t ; y ] in G G,E are equal if and only if x = y andthere exists an idempotent 0 = e ∈ S G,E such that e · x = x and se = te .The unit space of G G,E is G (0) G,E = { [ α, G , α ; x ] : x = α ˆ x } , HOSSEIN LARKI which is identified with E ∞ by [ α, G , α ; x ] x . Then, the range and sourcemaps are defined by r ([ α, g, β ; β ˆ x ]) = α ( g ˆ x ) and s ([ α, g, β ; β ˆ x ]) = β ˆ x. Following [7, Section 10], we endow G G,E with the topology generated bycompact open bisections of the formΘ( α, g, β ; Z ( γ )) := { [ α, g, β ; y ] ∈ G G,E : y ∈ Z ( γ ) } where γ ∈ E ∗ and Z ( γ ) := { γx : x ∈ s ( γ ) E ∞ } . Hence, G G,E is an amplegroupoid.
Definition . We say that (
G, E, ϕ ) is pseudo free if for every g ∈ G and e ∈ E , ge = e and ϕ ( g, e ) = 1 G = ⇒ g = 1 G . In the end of this section, we recall briefly the following results from [7]for convenience. Although they are proved there for finite self-similar graphswith no sources, but we can obtain them for countable cases by a same way(see also [8]).
Proposition 2.7.
Let ( G, E, ϕ ) be a pseudo free self-similar graphs withoutsources and let G G,E be the associated groupoid as above. Then (1) G G,E ∼ = G tight ( S G,E ) [7, Theorem 8.19] , G G,E is Hausdorff [7, Propo-sition 12.1] , and O G,E ∼ = C ∗ ( G G,E ) [7, Theorem 9.6] . (2) If moreover G is an amenable group, then G G,E is an amenablegroupoid in the sense of [1] . In particular, we have O G,E ∼ = C ∗ ( G G,E ) ∼ = C ∗ r ( G G,E ) by [1, Proposition 6.1.8] . Proposition 2.8 ([7, Corollary 14.13] and [8, Theorem 4.4]) . Let ( G, E, ϕ ) be a pseudo free self-similar graph with no sources. Then G G,E is effective if and only if the following properties hold: (1) Every G -circuit in E has an entry, and (2) for every v ∈ E and G = g ∈ G , the action of g on Z ( v ) isnontrivial (i.e., there is x ∈ Z ( v ) such that g.x = x ). Purely infinite self-similar graph C ∗ -algebras In [7, Corollary 16.3] and [8, Corollary 4.7], it is shown that when O G,E is simple and (
G, E, ϕ ) contains a G -circuit, then O G,E is purely infinite.In this section, we study purely infinite C ∗ -algebras O G,E of countable self-similar graphs in the sense of [25] without the simplicity assumption. Ourmain result here is a generalization of [7, Theorem 16.2] to countable self-similar graphs. Note that there is another well-known notion of pure in-finiteness from [11] which is equivalent to that of [25] for the simple cases.Moreover, our results in this section may be generalized for the Kirchberg-Rørdam’s notion using [11, Corollary 3.15] and the ideal structure [14, Corol-lary 6.15]. Note that the ‘effective’ property of groupoids is called essentially principal in [7, 8].
DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 7 Theorem 3.1.
Let ( G, E, ϕ ) be a pseudo free self-similar graph over anamenable group G . Suppose also that ( G, E, ϕ ) satisfies conditions (1) and(2) of Proposition 2.8 (i.e., the groupoid G G,E is effective). Then O G,E ispurely infinite if and only if every vertex projection s v is infinite in O G,E .Proof.
We must prove the “if” implication only. So suppose that for every v ∈ E , s v is infinite in O G,E . Let G = G G,E be the groupoid associated to(
G, E, ϕ ). By Proposition 2.7(2), G is amenable, so C ∗ r ( G ) = C ∗ ( G ) = O G,E .We know that the cylinders { Z ( α ) : α ∈ E ∗ } is a basis of compact open setsfor the topology induced on E ∞ = G (0) . Moreover, Proposition 2.8 says that G is effective. Hence, Proposition 2.3 implies that O G,E = C ∗ r ( G ) is purelyinfinite if and only if { Z ( α ) = s α s ∗ α : α ∈ E ∗ } are all infinite projections in O G,E . Now since s α s ∗ α ∼ s ∗ α s α = s d ( α ) and the infiniteness passes throughMurray-von Neumann equivalence, we conclude the result. (cid:3) Definition . Let v, w ∈ E . We say that v receives a G -path from w or w connects to v by a G -path , say v & w , if there exist α ∈ E ∗ and g ∈ G suchthat r ( α ) = v and d ( α ) = gw . By [7, Proposition 13.2], this is equivalent to ∃ α ∈ E ∗ , ∃ g ∈ G such that r ( α ) = gv and d ( α ) = w. Lemma 3.3.
Let ( G, E, ϕ ) be a self-similar graph. For v, w ∈ E and α, β ∈ E ∗ , we have (1) If v = gw for some g ∈ G , then s v ∼ s w in the Murray-von Neumannsense. (2) If v receives a G -path from w , then s v % s w . (3) If β = gα for some g ∈ G , then s β s ∗ β ∼ s α s ∗ α .Proof. (1). If v = gw , then we have s v = ( u g s v ) ∗ ( u g s v ) and( u g s v )( u g s v ) ∗ = ( s gv u g )( s gv u g ) ∗ = s w u g u ∗ g s w = s w , concluding s v ∼ s w .For (2), suppose that there exist α ∈ E ∗ and g ∈ G such that r ( α ) = v and d ( α ) = gw . Then, by the Cuntz-Krieger relations, s v ≥ s α s ∗ α ∼ s ∗ α s α = s d ( α ) = s gw ∼ s w , and consequently s v % s w .For (3), if β = gα , then by part (1) we have s β s ∗ β ∼ s ∗ β s β = s d ( β ) = s g.d ( α ) ∼ s d ( α ) = s ∗ α s α ∼ s α s ∗ α , giving s β s ∗ β ∼ s α s ∗ α . (cid:3) Proposition 3.4.
Let ( G, E, ϕ ) be a pseudo free self-similar graph over anamenable group G . Suppose that conditions (1) and (2) of Proposition 2.8hold. Then (1) If every v ∈ E receives a G -path from a G -circuit, then O G,E ispurely infinite. (2)
If the graph C ∗ -algebra C ∗ ( E ) is purely infinite, then so is O G,E . HOSSEIN LARKI
Proof. (1). In view of Theorem 3.1, it suffices to prove that each s v isinfinite in O G,E . So, fix some v ∈ E . By hypothesis, there is a G -circuit α connecting to v by a G -path.We first show that s r ( α ) is infinite. For, let γ be an entry for α by as-sumption. Since each of α nor γ is not a subpath of the other, one maycompute that s α s ∗ α and s γ s ∗ γ are orthogonal. Hence, the Cuntz-Krieger re-lations imply that s r ( α ) ≥ s α s ∗ α + s γ s ∗ γ > s α s ∗ α ∼ s ∗ α s α = s d ( α ) . If d ( α ) = gr ( α ), then s d ( α ) ∼ s r ( α ) by Lemma 3.3(1), and whence s r ( α ) isinfinite in O G,E as claimed.Now, because there is a G -path from r ( α ) to v , we have s v % s r ( α ) byLemma 3.3(2), and therefore s v is infinite as well. As v ∈ E was arbitrary,Theorem 3.1 follows the result.(2). If C ∗ ( E ) is purely infinite, then each s v is infinite in C ∗ ( E ), and sois in O G,E as well. Now apply Theorem 3.1. (cid:3)
Remark . If v ∈ E receives a G -path from a G -circuit with an entry butnot a path from a circuit, then s v is infinite in O G,E while not in C ∗ ( E ).Therefore, the converse of Proposition 3.4(2) does not necessarily hold.In the simple case we conclude the following. Corollary 3.6.
Let ( G, E, ϕ ) be a pseudo free self-similar graph over anamenable group G . Suppose that O G,E is simple. If E contains a G -circuit,then O G,E is purely infinite.Proof.
Note that the simplicity of O G,E gives conditions (1) and (2) inProposition 2.8 [8, Theorem 4.5]. So, by Theorem 3.1, it suffices to showthat s v is infinite for each v ∈ E .Let ( g, α ) be a G -circuit in E . By [7, Theorem 16.1], ( g, α ) has an entry,hence s r ( α ) is infinite as seen in the proof of Proposition 3.4(1).Fix an arbitrary v ∈ E . We may form the infinite path α ∞ = α ( gα )( g α ) · · · ,which is well-defined because d ( g n α ) = g n d ( α ) = g n gr ( α ) = r ( g n +1 α ) . Since E is also weakly G -transitive by [8, Theorem 4.5], there is a G -pathfrom r ( g n α ) to v for sufficiently large n . Note that as r ( g n α ) = g n r ( α ), s r ( g n α ) = s g n r ( α ) is infinite by Lemma 3.3(1). Also, Lemma 3.3(2) impliesthat s v % s r ( g n α ) ∼ s r ( α ) , and consequently s v is infinite too. As v ∈ E wasarbitrary, Theorem 3.1 concludes that O G,E is purely infinite. (cid:3)
Remark . The converse of above corollary will be proved in Theorem 4.9(1) ⇐⇒ (6).The following result gives necessary and sufficient criteria for the purelyinfinite simple C ∗ -algebras by the monoiod of equivalent projections. It isnew even for the ordinary graph C ∗ -algebras. Before that we recall the defi-nition of K -group of a unital C ∗ -algebra and establish some notations. Let DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 9 A be a unital C ∗ -algebra and write by P ( A ) the collection of all projectionsin M ∞ ( A ) = S n ≥ M n ( A ). We say that two projections p ∈ M m ( A ) and q ∈ M n ( A ) are equivalent, denoted by p ∼ q , if ∃ v ∈ M m,n ( A ) such that p = v ∗ v and q = v ∗ v. Note that, if m ≤ n , then p ∼ q if and only if p ⊕ n − m is Murray-vonNeumann equivalent to q in M n ( A ), where x ⊕ y := diag( x, y ). Define D ( A ) := P ( A ) / ∼ = { [ p ] : p ∈ P ( A ) } , which is an abelian monoid with theoperation [ p ] + [ q ] := [ p ⊕ q ]. Then K ( A ) is the Grothendieck group of D ( A )endowed with a universal Grothendieck map φ : D ( A ) → K ( A ). The imageof D ( A ) under φ is denoted by K ( A ) + . It is known that when D ( A ) \ { } is a group, then K ( A ) = D ( A ) \ { } . Theorem 3.8. (1)
Let E be an arbitrary directed graph (non necessarilyrow-finite, source-free, or even countable) with | E | < ∞ . Then C ∗ ( E ) is purely infinite and simple if and only if D ( C ∗ ( E )) \ { } isa group (or equivalently, D ( C ∗ ( E )) \ { } = K ( C ∗ ( E )) ). (2) Let ( G, E, ϕ ) be a pseudo free self-similar graph over an amenablegroup G . Suppose also that | E | < ∞ and conditions (1) and (2)of Proposition 2.8 hold. Then O G,E is purely infinite simple if andonly if D ( O G,E ) is a group.Proof. Note that the “only if” implications hold for every unital purely in-finite simple C ∗ -algebra. Indeed, if A is a purely infinite simple C ∗ -algebra,then nonzero projections of A are all infinite. Thus, combining Proposition1.5 and Theorem 1.4 of [5] implies that D ( A ) \ { } is a group (= K ( A )).So it is enough to prove the “if” parts. We first show that every projection p in A is infinite for any unital C ∗ -algebra A with D ( A ) \{ } a group. Indeed,if [ f ] is the identity of D ( A ) \ { } , then[ p ] = [ p ] + [ f ] = [ p ⊕ f ] , thus we have p ∼ p ⊕ < p ⊕ f ∼ p, where 0 is a zero matrix in M ∞ ( A ). Therefore, p is an infinite projection in A , as claimed.In the case of statement (1), this follows that E satisfies Condition (L).In fact if there exists a circuit in E with no entries, then C ∗ ( E ) contains anideal Morita equivalent to C ( T ), hence it has a finite projection. Recall thatby Condition (L) every ideal of C ∗ ( E ) has a (vertex) projection. Now takea nonzero ideal I of C ∗ ( E ) and some projection 0 = p ∈ I . As | E | < ∞ ,write 1 := P v ∈ E s v the unit of C ∗ ( E ). Then [ p ] + [1 − p ] = [1] and we have[ p ] = [1] + [ q ] = [1 ⊕ q ] , where [ q ] is the inverse of [1 − p ] in D ( C ∗ ( E )) \{ } . Therefore, p ∼ ⊕ q whichsays that there is x = [ x x x . . . ] ∈ M , ∞ ( O G,E ) such that x ∗ px = 1 ⊕ q .In particular, 1 = x ∗ px ∈ I , concluding I = C ∗ ( E ). Therefore C ∗ ( E ) issimple. For the pure infiniteness, let B be a nonzero hereditary C ∗ -subalgebra of C ∗ ( E ). Again, Condition (L) gives a nonzero projection p in B . If [ f ] is theidentity of D ( C ∗ ( E )) \ { } , then[ p ] = [ p ] + [ f ] = [ p ⊕ f ] , and we have p ∼ p ⊕ < p ⊕ f ∼ p where 0 is a zero matrix in M ∞ ( O G,E ), and consequently p is infinite. There-fore, C ∗ ( E ) is purely infinite.For statement (2), note that G G,E is effective by Proposition 2.8, and O G,E ∼ = C ∗ r ( G G,E ) by Proposition 2.7. This implies that every ideal of O G,E contains a projection (see [6, Theorem4.4] for example). Now we may followthe proof of statement (1) to obtain the result. (cid:3) Stable finiteness and a dichotomy
In this section, we associate a special graph e E to any self-similar graph( G, E, ϕ ). We show that if the graph C ∗ -algebra C ∗ ( e E ) is either simple,purely infinite, or stable finite then so is O G,E respectively. Then we willconclude a dichotomy for simple self-similar graph C ∗ -algebras. Definition . Let K denote the C ∗ -algebra of compact operators on aseparable, infinite dimensional Hilbert space. A (simple) C ∗ -algebra A iscalled stably finite if A ⊗ K contains no infinite projections.Fix a self-similar graph ( G, E, ϕ ). In the following we define a graph e E associated to ( G, E, ϕ ). Define ≈ on E ∗ = F ∞ n =0 E n by α ≈ β ⇐⇒ ∃ g ∈ G such that β = gα, which is an equivalent relation on each E n (and so on E ∗ ). The vertexset of e E is e E := E / ≈ the collection of vertex classes. In each class[ v ] ∈ e E pick exactly one vertex up and collect them in the set Ω. Hence, e E = { [ v ] : v ∈ Ω } , and we have [ v ] = [ w ] for v = w ∈ Ω. For every v ∈ Ωand e ∈ r − ( v ) draw an edge ˜ e from [ d ( e )] to [ v ]. Hence we obtain the graph e E so that e E := { [ v ] : v ∈ Ω } , and e E := [ v ∈ Ω ^ r − ( v ) = [ v ∈ Ω { e e : r ( e ) = v } , with the range e r ( e e ) = [ r ( e )] and domain e d ( e e ) = [ d ( e )] for every e e ∈ e E . Example . For n ≥
1, let Z mod n be the additive group { , , . . . , n } . Let( Z mod n , E, ϕ ) be a triple with the cyclic graph E DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 11 vw w w w n f f n g n ······ e e e n g f g and the action Z mod n y E defined by kv := v and kα i := α k + i (1 ≤ k, i ≤ n ) , for every α i ∈ { w i , e i , f i , g i } . Since w i ≈ w j , for any 1 ≤ i, j ≤ n , we mayselect w of the class [ w ] = { w , . . . , w n } . As r − ( v ) = { e , . . . , e n } and r − ( w ) = { f , g n } , then the graph e E would be[ v ][ w ] · · · e e e e e e n e f e g n Lemma 4.3.
Let ( G, E, ϕ ) be a self-similar graph, and consider an associ-ated graph e E as above. Then (1) If E is row-finite, then so is e E . (2) For each finite path e α = e α . . . e α n ∈ e E n , there is a path γ = γ . . . γ n in E n such that γ ≈ α i for ≤ i ≤ n . Conversely, if γ = γ . . . γ n ∈ E n , then there exists e α = e α . . . e α n ∈ e E n such that γ ≈ α i for ≤ i ≤ n . (3) If e α ∈ e E n and γ ∈ E n are two paths as in statement (2), then e α isa circuit in e E if and only if γ is a G -circuit in E . Moreover, e α hasan entry if and only if γ does.Proof. Statement (1) is clear by the definition of e E . For (2), let first e α = e α . . . e α n ∈ e E n be a path in e E . Then, for each 1 ≤ i < n , we have[ d ( α i )] = e d ( e α i ) = e r ( e α i +1 ) = [ r ( α i +1 )] , and so there exists g i ∈ G such that d ( α i ) = g i r ( α i +1 ). Now set γ := α and γ i := g . . . g i − α i for every 2 ≤ i ≤ n . Then d ( γ i ) = d ( g . . . g i − α i ) = g . . . g i − d ( α i ) = g . . . g i − g i r ( α i +1 ) = r ( γ i +1 ) , and hence γ = γ . . . γ n is a desired path in E . Conversely, let γ = γ . . . γ n be a finite path in E n . For each 1 ≤ i ≤ n ,there is v i ∈ Ω such that v i = g i r ( γ i ) for some g i ∈ G . Hence, we have e α = ( g g γ ) . . . ( ] g n γ n ) ∈ e E with α ≈ γ .For statement (3), given e α and γ as in part (2), we have e α is a circuit in E ⇐⇒ [ d ( α n )] = [ r ( α )] ⇐⇒ d ( α n ) ≈ r ( α ) ⇐⇒ d ( γ n ) ≈ d ( α n ) ≈ r ( α ) ≈ r ( γ ) ⇐⇒ γ is a G − circuit . Moreover, since | r − ( r ( γ i )) | = | e r − ( e r ( e α i )) | for each 1 ≤ i ≤ n , we have γ has an entry ⇐⇒ | r − ( r ( γ i )) | > ≤ i ≤ n ⇐⇒ | e r − ( e r ( α i )) | > ≤ i ≤ n ⇐⇒ e α has an entry in e E. (cid:3) Definition . Let (
G, E, ϕ ) be a self-similar graph. Following [7, Definition3.4], we say that E is weakly G -transitive if for every v ∈ E and x ∈ E ∞ ,there exists a path α such that d ( α ) = x ( n, n ) for some n ≥ r ( α ) = gv for some g ∈ G . If we have an ordinary graph E (with the trivial groupaction), we say simply that E is weakly transitive . Note that the weaklytransitive is called cofinal in [20]. Lemma 4.5.
Let ( G, E, ϕ ) be a self-similar graph, and associate a graph e E as above. Then (1) Every G -circuit in E has an entry if and only if every circuit in e E does. (2) E is weakly G -transitive if and only if e E is weakly transitive.Proof. Statement (1) follows from items (2) and (3) of Lemma 4.3. For (2),let e E be transitive. Take an arbitrary infinite path x ∈ E ∞ and some v ∈ E .By item (2) in Lemma 4.3, there is e y ∈ e E ∞ such that y (0 , n ) ≈ x (0 , n ) forevery n ≥
0. By transitivity, there exists e γ ∈ e E ∗ such that e r ( e γ ) = [ v ] and e d ( e γ ) = [ y ( n, n )] for some n . Hence, v ≈ r ( γ ) and d ( γ ) ≈ y ( n, n ) ≈ x ( n, n ).This follows that E is G -transitive. The converse is analogous. (cid:3) Proposition 4.6.
Let ( G, E, ϕ ) be a self-similar graph over an amenablegroup G , and let e E be an associated graph. (1) In case the groupoid G G,E is Hausdorff (see [8, Theorem 4.2] ), then O G,E is simple if and only if (a) the graph C ∗ -algebra C ∗ ( e E ) is simple, and (b) for v ∈ E and g ∈ G , if the action of g on the cylinder Z ( v ) is trivial (i.e., gx = x for every x ∈ Z ( v ) ), then g is slack at v . DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 13 (2) Suppose that ( G, E, ϕ ) is pseudo free and for any v ∈ E and G = g ∈ G , the action of g on Z ( v ) is nontrivial. If C ∗ ( e E ) is purelyinfinite, then so is O G,E .Proof.
Statement (1) follows from Lemma 4.5 and [8, Theorem 4.5]. For(2), if the graph C ∗ -algebra C ∗ ( e E ) is purely infinite, then every circuit in e E has an entry and every vertex [ v ] ∈ e E can be reached from a circuit.By Lemma 4.5, every G -circuit has an entry and every v ∈ E receives a G -path from a G -circuit. Now, Proposition 3.4(1) concludes that O G,E ispurely infinite. (cid:3)
Example . The graph e E in Example 4.2 is weakly transitive and everycircuit in e E has an entry. Then C ∗ ( e E ) is simple and purely infinite, and sois the C ∗ -algebra O G,E by Proposition 4.6.
Definition . Let (
G, E, ϕ ) be a self-similar graph. A graph trace on E is map T : E → R + such that(1) T ( r ( e )) ≥ T ( d ( e )) for every e ∈ E , and(2) T ( v ) = P r ( e )= v T ( d ( e )) for every v ∈ E .A graph G -trace in E is a graph trace T : E → R + such that T ( v ) = T ( w )for every v ≈ w in E . Theorem 4.9.
Let ( G, E, ϕ ) be a pseudo free self-similar graph over anamenable group G . Suppose that O G,E is simple. Then the following areequivalent. (1) O G,E is stably finite. (2) O G,E is quasi diagonal. (3) (
G, E, ϕ ) has a nonzero graph G -trace. (4) e E has a nonzero graph trace. (5) e E contains no circuits. (6) E contains no G -circuits.Proof. Statements (1) and (2) are equivalent by [21, Corollary 6.6].(1) ⇒ (6). If E has a G -circuit, then O G,E is purely infinite by Corollary3.6. In particular, O G,E is not stably finite, a contradiction.(6) ⇒ (5) follows from Lemma 4.3(3).(5) ⇒ (4). Suppose that e E has no circuits. Arrange e E = { [ v ] , [ v ] , . . . } .For each n ≥
1, let F n be the full subgraph of e E containing all S ni =1 e r − ([ v i ]).Since F n ’s have no circuits, [13, Corollary 2.3] implies that C ∗ ( F ) ⊆ C ∗ ( F ) ⊆ . . . is a sequence of finite dimensional C ∗ -subalgebras of C ∗ ( e E ) such that C ∗ ( e E ) = lim C ∗ ( F n ) (i.e., C ∗ ( e E ) is AF). Thus there exist bounded traces τ n : C ∗ ( F n ) → C such that τ n | C ∗ ( F i ) equals with τ i for i ≤ n . This inducesa semifinite trace τ = lim τ n on C ∗ ( e E ). Therefore, if C ∗ ( e E ) = C ∗ ( t e , q [ v ] ),we obtain the nonzero graph trace T : e E → R + , by T ([ v ]) = τ ( q [ v ] ), on e E .(4) ⇒ (3). Suppose that T is a nonzero graph trace on e E . Note that,since the action of G on E gives automorphisms respecting to the range and domain, for any v = w ∈ E with w = gv , the map e ge is a bijectionfrom r − ( v ) onto r − ( w ). In particular, | r − ( w ) | = | r − ( v ) | . Being thisfact in mind, one may easily see that the map T ′ : E → R + , defined by T ′ ( v ) := T ([ v ]), is a nonzero graph G -trace on E , as desired.(3) ⇒ (1). By [23, Proposition II.4.8], there exists a faithful conditionalexpectation π : C ∗ ( G G,E ) → C ( G (0) G,E ) such that π ( f ) = f | G (0) G,E for all f ∈ C c ( G (0) G,E ). Note that the isomorphism ψ : O G,E → C ∗ ( G G,E ) in Proposition2.7(1) maps the core O G,E := span { s α s ∗ α : α ∈ E ∗ } onto C ( G (0) G,E ). Hence ϕ := ψ − ◦ π ◦ ψ is a faithful conditional expectation from O G,E onto O G,E such that φ ( s α u g s ∗ β ) = (cid:26) s α s ∗ α β = α, g = 1 G α, β ∈ E ∗ and g ∈ G .Now suppose that T is a nonzero graph G -trace on E . Define t : O G,E → C by t ( s α s ∗ α ) = T ( d ( α )), which is a linear functional on O G,E . So, wemay easily verify that τ := t ◦ φ is a semifinite trace on O G,E such that0 < τ ( s v ) < ∞ for all v ∈ E . Moreover, τ is faithful because O G,E issimple. Thus [21, Corollary 6.6] yields that O G,E is stably finite. (cid:3)
Recall from [7, Corollary 10.16] that if G is amenable, then O G,E is anuclear C ∗ -algebra. So, combining Corollary 3.6 and Theorem 4.9 impliesthe following dichotomy for simple O G,E . Corollary 4.10.
Let ( G, E, ϕ ) be a pseudo free self-similar graph over anamenable group G . Suppose that O G,E is simple. Then (1) If E has a G -circuit, then O G,E is purely infinite. In this case, O G,E is a Kirchberg algebra, and we have K ( O G,E ) = D ( O G,E ) \ { } whenever | E | < ∞ . (2) Otherwise, O G,E is stably finite. In this case, ( K ( O G,E ) , K ( O G,E ) + ) is an ordered abelian group (see [24, Proposition 5.1.5(iv)] ).Remark . Note that in case O G,E is stably finite, the embedding ι : C ∗ ( E ) ֒ → O G,E of [7, Section 11] induces an embedding K ( ι ) : K ( C ∗ ( E )) ֒ → K ( O G,E ) defined by K ( ι )([ p ] ) := [ ι ( p )] , where the map ι is naturally ex-tended on M ∞ ( C ∗ ( E )) into M ∞ ( O G,E ). Indeed, if p ∈ M ∞ ( C ∗ ( E )) is aprojection with [ ι ( p )] = 0, then we must have ι ( p ) = 0 because M ∞ ( O G,E )has no infinite projection, and hence p = 0.5. Pure infiniteness of self-similar k -graph C ∗ -algebras In this section, we consider the pure infiniteness of self-similar k -graph C ∗ -algebras. Let us first recall the definitions of self-similar k -graphs andtheir C ∗ -algebras from [17]. Fix k ∈ N ∪ {∞} and let Λ = (Λ , Λ , r, s )be a row-finite k -graph with no sources (we refer the reader to [20] forbasic definitions and concepts about k -graphs and associated C ∗ -algebras). DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 15 Consider N k as a category with a single object 0 and the coordinatewisepartial order ≤ . Let Ω k := { ( p, q ) : p, q ∈ N k , p ≤ q } . An infinite path in Λis a morphism x : Ω k → Λ with the range r ( x ) := x (0 , ∞ the set of infinite paths in Λ.Let G be a (discrete and countable) group. An action G y Λ is a map G × Λ → Λ, ( g, λ ) → gλ , which gives a graph automorphism preserving thedegree map for every g ∈ G . Definition . A self-similar k -graph is a triple ( G, Λ , ϕ ), where Λ isa k -graph, G is a group acting on Λ, and ϕ : G × Λ → Λ is a cocycle for G y Λ with the property ϕ ( g, λ ) .v = gv ( g ∈ G, v ∈ Λ , λ ∈ Λ) . Following [17], we consider only self-similar k -graphs ( G, Λ , ϕ ) for row-finite and source-free k -graphs with | Λ | < ∞ . We will write ( G, Λ , ϕ ) by( G, Λ) for simplicity. Note that ϕ was called the restriction map in [17] andeach ϕ ( g, λ ) was denoted by g | λ there. Definition . Let ( G, Λ) be a self-similar k -graph. We say that(1) ( G, Λ) is pseudo free , if gλ = λ and ϕ ( g, λ ) = 1 G imply g = 1 G .(2) ( G, Λ) is G -aperiodic if for any v ∈ Λ , there exists x ∈ v Λ ∞ suchthat x ( p, ∞ ) = gx ( q, ∞ ) implies g = 1 G and p = q for p, q ∈ N k and g ∈ G .(3) ( G, Λ) is G -cofinal if for every x ∈ Λ ∞ and v ∈ Λ , there exist p ∈ N k , µ ∈ Λ, and g ∈ G such that s ( µ ) = x ( p, p ) and r ( µ ) = gv . Definition . Let ( G, Λ) be a self-similar k -graph as in Definition 5.1 with | Λ | < ∞ . The C ∗ -algebra O G, Λ associated to ( G, Λ) is the universal C ∗ -algebra generated by { s λ : λ ∈ Λ } and { u g : g ∈ G } such that(1) { s λ : λ ∈ Λ } is a Cuntz-Krieger Λ-family in the sense of [12].(2) u : G → O G, Λ , given by g u g , is a unitary ∗ -representation of G .(3) u g s λ = s gλ u ϕ ( g,λ ) for every g ∈ G and λ ∈ Λ.Similar to the construction of G G,E in Section 2.4, Li and Yang associatedan ample groupoid G G, Λ in [17, Section 5.1] such that O G, Λ ∼ = C ∗ ( G G, Λ ) ∼ = C ∗ r ( G G, Λ ) when G is amenable and ( G, Λ) is pseudo free [17, Theorem 5.9].In particular, the unit space G (0) G, Λ is homeomorphic to Λ ∞ endowed with thetopology generated by cylinders Z ( λ ) := { λx : x ∈ Λ ∞ } .Recall that a circuit in Λ is a path α ∈ Λ with r ( α ) = s ( α ). τ ∈ Λ iscalled an entry for α if r ( τ ) = r ( α ) and there are no common extensions for α and τ (i.e., αµ = τ ν for all µ, ν ∈ Λ).
Theorem 5.4.
Let ( G, Λ) be a pseudo free self-similar k -graph with | Λ | < ∞ over an amenable group G . If Λ is G -aperiodic, then O G, Λ is purelyinfinite. In particular, if Λ is also G -cofinal, then O G, Λ is a Kirchbergalgebra. Proof.
Let G G, Λ be the groupoid associated to ( G, Λ). Then G G, Λ is amenableand effective [17, Proposition 6.5], and we thus have C ∗ ( G G, Λ ) = C ∗ r ( G G, Λ ) = O G, Λ by [17, Theorem 5.9]. We know that the cylinders { Z ( λ ) : λ ∈ Λ } forma basis of compact open sets for the topology on Λ ∞ = G (0) G, Λ . So, in light ofProposition 2.3, it suffices to prove that each 1 Z ( λ ) is an infinite projectionfor λ ∈ Λ. For this, since1 Z ( λ ) = s λ s ∗ λ ∼ s ∗ λ s λ = s s ( λ ) , we show all s v ’s are infinite in O G, Λ for v ∈ Λ .So fix an arbitrary v ∈ Λ . We claim that v reaches from a circuitwith an entry. To see this, take some x ∈ v Λ ∞ . For any t ∈ N , write t := ( t, , , . . . ) ∈ N k . Since { x ( t , t ) : t ≥ } ⊆ Λ is finite, there are t < t such that x ( t , t ) = x ( t , t ). Hence x ( t , t ) is a circuit in Λ,which connects to v by x (0 , t ) ∈ Λ. Note that the G -aperiodicity yieldsclearly the periodicity of Λ. Hence, one may follow [16, Lemma 6.1] to findan (initial) circuit α with an entry τ connecting to v , as claimed.Since α and τ have no common extensions, one may compute that s α s ∗ α and s τ s ∗ τ are orthogonal (by applying [20, Lemma 9.4]). Thus, by the Cuntz-Krieger relations we have s r ( α ) ≥ s α s ∗ α + s τ s ∗ τ > s α s ∗ α ∼ s ∗ α s α = s s ( α ) = s r ( α ) , so s r ( α ) is infinite. Moreover, if λ connects r ( α ) to v , then s v ≥ s λ s ∗ λ ∼ s ∗ λ s λ = s s ( λ ) = s r ( α ) , which says that s v is an infinite projection in O G, Λ as well. Since v ∈ Λ was arbitrary, this deduces that O G, Λ is purely infinite by Proposition 2.3.For the last statement, if moreover Λ is G -cofinal, then [17, Theorem 6.6]implies that O G, Λ is nuclear and simple, which satisfies UCT. Hence, O G, Λ is a Kirchberg algebra. (cid:3) Corollary 5.5 (See [17, Theorem 6.13]) . Let ( G, Λ) be a pseudo free self-similar k -graph with | Λ | < ∞ over an amenable group G . Whenever O G, Λ is simple, then it is purely infinite too. References [1] C. Anantharaman-Delaroche and J. Renault.
Amenable groupoids , volume 36 ofMonographs of L’Enseignement Mathmatique, Geneva, 2000.[2] E. B´edos, S. Kaliszewski, J. Quigg,
On Exel-Pardo algebras , J. Operator Theory (2)(2017), 309-345.[3] C. B¨onicke and K. Li, Ideal structure and pure infiniteness of ample groupoid C ∗ -algebras , Ergodic Theory Dynam. Systems (2018), 1-30. doi:10.1017/etds.2018.39[4] J.H. Brown, L.O. Clark and A. Sierakowski, Purely infinite C ∗ -algebras associated to´etale groupoids , Ergodic Theory Dynam. Systems (2015), 2397-2411.[5] J. Cuntz. K -theory for certain C ∗ -algebras , Ann. Math. (1981), 181-197.[6] R. Exel, Non-Hausdorff ´ e tale groupoids , Proc. Amer. Math. Soc. (2011), 897-907.[7] R. Exel, E. Pardo, Self-similar graphs, a unified treatment of Katsura and Nekra-shevych C ∗ -algebras , Adv. Math. (2017), 1046-1129. DICHOTOMY FOR SIMPLE SELF-SIMILAR GRAPH C ∗ -ALGEBRAS 17 [8] R. Exel, E. Pardo and C. Starling, C ∗ -algebras of self-similar graphs over arbitrarygraphs , preprint, arXiv:1807.01686 (2018).[9] J. Hjelmborg, Purely infinite and stable C ∗ -algebras of graphs and dynamical systems ,Ergodic Theory Dynam. Systems (2001), 1789-1808.[10] T. Katsura, A construction of actions on Kirchberg algebras which induce given ac-tions on their K -groups , J. Reine Angew. Math. (2008), 27-65.[11] E. Kirchberg and M. Rørdam, Non-simple purely infinite C ∗ -algebras , Amer. J. Math. (2000), 637-666.[12] A. Kumjian and D. Pask, Higher rank graph C ∗ -algebras , New York J. Math. (2000),1-20.[13] A. Kumjian, D. Pask and I. Raeburn, Cuntz-Krieger algebras of directed graphs ,Pacific J. Math. (1998), 161-174.[14] S.M. LaLonde, D. Milan, and J. Scott,
Condition (K) for inverse semigroups and theideal structure of their C ∗ -algebras , J. Algebra (2019), 119-153.[15] H. Larki, Non-simple purely infinite Steinberg Algebras with applications to Kumjian-Pask algebras , preprint, arXiv:1901.07094 (2019)[16] H. Larki,
Purely infinite simple Kumjian-Pask algebras , Forum Math. (1) (2018),253-268.[17] H. Li and D. Yang, Self-similar k -graph C ∗ -algebras , preprint, arXiv:1712.08194(2018).[18] V. Nekrashevych, Cuntz-Pimsner algebras of group actions , J. Operator Theory (2004), 223-249.[19] V. Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs,vol.117, Amer. Math. Soc., Providence RI 2005.[20] I. Raeburn, Graph Algebras, CBMS Regional Conf. Ser. in Math., vol. 103, Amer.Math. Soc., Providence RI 2005.[21] T. Rainone and A. Sims, A dichotomy for groupoid C ∗ -algebras , Ergod. Th. Dynam.Sys. (2018), 1-43, doi:10.1017/etds.2018.52.[22] J. Renault, Cartan subalgebras in C ∗ -algebras , Irish Math. Soc. Bulletin (2008),2963.[23] J. Renault, A groupoid approach to C ∗ -algebras, Lecture Notes in Mathematics, vol.793, Springer, Berlin, 1980.[24] M. Rørdam, F. Larsen and N. Laustsen, An introduction to K -theory for C ∗ -algebras,London Mathematical Society Student Texts 49, Cambridge University Press, Cam-bridge, 2000.[25] M. Rørdam and E. Størmer, Classification of nuclear C ∗ -algebras Entropy in operatoralgebras, Operator Algebras and Non-commutative Geometry, 7, Springer-Verlag,Berlin, 2002. Department of Mathematics, Faculty of Mathematical Sciences and Com-puter, Shahid Chamran University of Ahvaz, P.O. Box: 83151-61357, Ahvaz,Iran
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