On the Simplicity of C ∗ -algebras Associated to Multispinal Groups
aa r X i v : . [ m a t h . OA ] F e b ON THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINALGROUPS KEISUKE YOSHIDA
Abstract.
We characterize the simplicity of C ∗ -algebras arising from multispinal groups. Let O G max be the universal C ∗ -algebra associated to a multispinal group G . We show that the invertibility of amatrix completely determines the simplicity of O G max . Introduction
Many finitely generated C ∗ -algebras have been studied since the first half of 20th century whenoperator algebras were introduced by F. J. Murray and J. von Neumann. Some of them are constructedfrom dynamical systems. Such C ∗ -algebras include the Cuntz algebras [11]. The Cuntz algebra O n can be regarded as the C ∗ -algebra arising from the shift maps on the symbolic dynamical system overa finite set X with | X | = n . Here n ≥ X ω for the set of right infinitewords over X . For each x ∈ X , the shift map T x : X ω → X ω is given by T x ( w ) := xw for w ∈ X ω . One often identifies each shift map with an isometry on a Hilbert space. In this paper, wewrite S x for T x when we regard T x as an operator. Recall that the Cuntz algebra O n is the universalC ∗ -algebra generated by { S x } x ∈ X satisfying(1.1) S ∗ x S x = 1 , X y ∈ X S y S ∗ y = 1for any x ∈ X .We write Homeo( X ω ) for the group of homeomorphisms on X ω . A subgroup G ⊂ Homeo( X ω ) issaid to be a self-similar group over X if for any g ∈ G, x ∈ X there exist g | x ∈ G, g ( x ) ∈ X with(1.2) gS x = S g ( x ) g | x . Let µ be the Bernoulli measure on X ω and let ( X ω ) g be the set of fixed points of g ∈ G .Self-similar groups play important rolls in geometric group theory. For instance, a self-similargroup called the Grigorchuk group is known as the first example of a finitely generated group ofintermediate growth [13]. The Grigorchuk group is a self-similar group over { , } . Iterated monodromygroups are also important examples. They provide useful techniques to the study of some complexdynamical systems (see [3] for details). In addition, the following operator algebraic approach toiterated monogromy groups gives a K -theoretic invariant for complex dynamical systems (see [18]).We study C ∗ -algebras arising from self-similar groups. V. V. Nekrashevych has considered ∗ -representations of self-similar groups. In [18], he introduced the universal C ∗ -algebra generated bya self-similar group G and { S x } x ∈ X satisfying equations (1.1), (1.2). In this paper, we write O G max forthe universal C ∗ -algebra. We often assume that G has a finite generation property called contracting(see Definition 2.10). Then O G max is finitely generated [18]. The C ∗ -algebra O G max is the main objectof this paper. Especially, we discuss the simplicity of O G max . Date : February 12, 2021.2020
Mathematics Subject Classification.
Key words and phrases. self-similar group, multispinal group, KMS state, groupoid C ∗ -algebra. One can construct the groupoid [
G, X ] of germs from self-similar group G over X . In [18], it wasshown that the universal C ∗ -algebra O G max is isomorphic to the full groupoid C ∗ -algebra C ∗ ([ G, X ])(see [12] for more general case). The groupoid [
G, X ] is minimal, effective and ample. Thus, if thegroupoid [
G, X ] is Hausdorff and amenable, then the C ∗ -algebras C ∗ ([ G, X ]) and O G max are simple.However, there exists a self-similar group whose groupoid of germs is not Hausdorff. Such self-similargroups include the Grigorchuk group. In [8], L. O. Clark et al . characterized the simplicity of C ∗ -algebras of non-Hausdorff groupoids. Using the characterization, they show O G max is simple if G is theGrigorchuk group. B. Steinberg and N. Szak´acs studied Steinberg algebras of non-Hausdorff groupoidsarising from self-similar groups in [20]. Combining results in [8, 20], we obtain self-similar groupswhich provide non-Hausdorff amenable ample minimal effective groupoids with non-simple reduced(and full) groupoid C ∗ -algebras. Some of such self-similar groups are multispinal groups [20]. Amultispinal group over X is a self-similar group arising from two finite groups A, B and a map Ψfrom X to Aut( A ) ∪ Hom(
A, B ). Here Aut( A ) is the set of automorphisms on A and Hom( A, B ) isthe set of homomorphisms from A to B . We do not observe the definition here, see section 3 for it.Note that every multispinal group is contracting [20]. In addition, the groupoid [ G, X ] of germs istypically non-Hausdorff if G is a multispinal group over X . In [21], Z. ˘Suni´c constructed a familyof self-similar groups which produces a generalization of the Grigorchuk group. Multispinal groupsinclude this family. For instance, the Grigorchuk group is a multispinal group arising from two finitegroups Z × Z and Z . Multispinal groups include other interesting self-similar groups. For example,Gupta-Sidki p -groups [14], GGS-groups [4] and multi-edge spinal groups [2] are multispinal groups.See [20] for details.In this paper, we write O G min for a certain simple quotient of O G max . The quotient O G min is alsointroduced by V. V. Nekrashevych [17]. We write h G, X i := { S u gS ∗ v : u, v ∈ X ∗ , g ∈ G } . For a representation of O G min , we consider the following state. Define a function ψ on h G, X i by ψ ( S u gS ∗ v ) = δ u,v µ (( X ω ) g )Here δ is the Kronecker delta. If G is contracting, then the function ψ extends to KMS states ψ min and ψ max on O G min and O G max , respectively [7, 16, 23]. Here note that we have canonical embeddingsof G into O G min and O G max . The uniqueness of KMS states is studied in [7, 16, 23] but we do not needthe uniqueness in this paper.We discuss when the universal C ∗ -algebra O G max is simple under the assumption that G is a mul-tispinal group and [ G, X ] is amenable. Note that we do NOT assume [
G, X ] is Hausdorff. If [
G, X ]is amenable and C ∗ r ([ G, X ]) is isomorphic to O G min , then the universal C ∗ -algebra O G max is simple.To get an isomorphism between O G min and C ∗ r ([ G, X ]), we compare two Hilbert spaces. One of theHilbert spaces is the GNS space H ψ min of ψ min . The reduced C ∗ -algebra C ∗ r ([ G, X ]) acts on the otherone. For the comparison between two Hilbert spaces, we consider the full group C ∗ -algebra C ∗ ( A ) ofthe finite subgroup A . We identify C ∗ ( A ) with finite dimensional C ∗ -subalgebra of O G max generatedby A . As a consequence, we obtain the characterization of the simplicity as follows (see Theorem 3.13). Main Theorem.
Let G = G ( A, B, Ψ) be a multispinal group over X . Assume that [ G, X ] is amenable.Then the following conditions are equivalent. (i) The restriction of ψ max to C ∗ ( A ) is faithful. (ii) The | A | × | A | matrix [ ψ ( a − a )] a ,a ∈ A is invertible. (iii) The universal C ∗ -algebra O G max is simple. Under the assumption of the Main Theorem, if one of the equivalent conditions holds, then O G max is a Kirchberg algebra. In addition, one can compute ψ ( a ) = µ (( X ω ) a ) for each a ∈ A by solving alinear equation. It is not hard to check that the condition (ii) holds for the Grigorchuk group (seeExample 3.17). Thus, as a corollary of the Main Theorem, we obtain a different proof of [8, Theorem5.22] which claims that O G max is a Kirchberg algebra for the Grigorchuk group G . N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 3 Preliminaries
In this section, we recall the definition of self-similar groups and related topics.2.1.
Self-similar groups.
We begin with notations used in this paper.
Notation 2.1.
Let X be a finite set X with | X | ≥
2. In this paper, X ∗ denotes the set of finite wordsover the alphabet X . In other words, X ∗ = F n ∈ N X n (we define X := {∅} ). Write X ω for the set ofunilateral infinite words over X . For w ∈ X ω and n ∈ N , w ( n ) ∈ X n denotes the first n letters of w .For a finite word u ∈ X n , we write | u | for the length of the word u , namely | u | = n . Take a subset P ⊂ X ω . We define uP := { uw : w ∈ P } ⊂ X ω . We identify X N with X ω . Thus, it is equipped with the product topology of the discrete sets X . Notethat X ω is homeomorphic to the Cantor space. Write Homeo( X ω ) for the group of homeomorphismson X ω . Definition 2.2. ([17, Definition 2.1]) A subgroup G of Homeo( X ω ) is said to be a self-similar groupover X if for every g ∈ G and x ∈ X there exist h ∈ G and v ∈ X with(2.1) g ( xw ) = vh ( w )for any w ∈ X ω .In this paper, G always denotes a self-similar group over X where X is a finite set with at least 2elements. Remark 2.3.
Using the equation (2.1) several times, we see that for every n ∈ N , g ∈ G and u ∈ X n there exist h ∈ G and v ∈ X n with g ( uw ) = vh ( w )for any w ∈ X ω . A direct calculation shows that h and v are uniquely determined by g and u . Write h = g | u and v = g ( u ).The following finitely generated group was first introduced in [13]. Example 2.4. ([17, Example 2.5]) Let X = { , } . Consider homeomorphisms a, b, c, d on X ω givenby a (0 w ) = 1 w, a (1 w ) = 0 w,b (0 w ) = 0 a ( w ) , b (1 w ) = 1 c ( w ) ,c (0 w ) = 0 a ( w ) , c (1 w ) = 1 d ( w ) ,d (0 w ) = 0 w, d (1 w ) = 1 b ( w )for w ∈ X ω . Let G be the subgroup of Homeo( X ω ) generated by a, b, c, d . The above relations implythat G is a self-similar group. The group G is called the Grigorchuk group .For more details and examples, see [16, 18, 20]. We often keep the following regularity in our mindin the study of self-similar groups and their operator algebras.
Definition 2.5. ([17, Definition 4.1]) Let G be a self-similar group over X and fix g ∈ G . An element w ∈ X ω is said to be g -regular if either g ( w ) = w or there exists an open neighborhood of w consistingof fixed points of g . Let ( X ω ) g -reg be the set of all g -regular points. Write ( X ω ) G -reg := T g ∈ G ( X ω ) g -reg .An element w ∈ ( X ω ) G -reg is said to be G -regular .Recall notations. Take any u ∈ X ∗ and let T u be the shift map on X ω given by w uw . Let T ∗ u denote the local inverse map of T u defined on the range of T u . Write h G, X i := { T u gT ∗ v : u, v ∈ X ∗ , g ∈ G } . KEISUKE YOSHIDA
Definition 2.6. ([17, Definition 9.1]) Let G be a self-similar group over X and fix f ∈ h G, X i . Anelement w ∈ X ω is said to be f -regular if either f is not defined on w or does not fix w or there existsan open neighborhood of w consisting of fixed points of f . We write ( X ω ) f -reg for the set of f -regularpoints. Also write ( X ω ) S G -reg := T f ∈h G,X i ( X ω ) f -reg . An element in ( X ω ) S G -reg is said to be strictly G -regular .From now on, we always assume that G is a countable self-similar group. The following remark tellsus why we need the countability. Remark 2.7.
It is not hard to show that ( X ω ) f -reg is an open dense subset of X ω for any f ∈ h G, X i .Thus, the Baire category theorem implies that ( X ω ) S G -reg is dense in X ω by the countability of G . Notethat ( X ω ) S G -reg is a h G, X i -invariant set.For f ∈ h G, X i , let ( X ω ) f be the set of fixed points of f . By definition, X ω \ ( X ω ) f -reg ⊂ ( X ω ) f .Let µ be the product measure of the uniform probability measures on X ’s (we identify X ω with X N ).We often refer to the measure µ as the Bernoulli measure. For any u ∈ X ∗ , we have µ ( uX ω ) = | X | −| u | .We recall the following facts in [23]. Lemma 2.8. ([23, Lemma 2.10])
Let G be a countable self-similar group over X . Then the followingconditions are equivalent. (i) µ (( X ω ) G -reg ) = 1 . (ii) µ (( X ω ) g -reg ) = 1 for any g ∈ G . (iii) µ (( X ω ) S G -reg ) = 1 . (iv) µ (( X ω ) f -reg ) = 1 for any f ∈ h G, X i . Theorem 2.9. ([23, Theorem 2.12])
For any countable self-similar group G over X , one has either µ (( X ω ) G -reg ) = 1 or µ (( X ω ) G -reg ) = 0 . For an example of a self-similar group G over X with ( X ω ) G -reg = 1, we recall the following property. Definition 2.10. ([18, Definition 2.2]) A self-similar group G over X is said to be contracting if thereexists a finite set N ⊂ G with the following condition:For any g ∈ G there exists n ∈ N satisfying g | v ∈ N for any v ∈ X ∗ with | v | > n .If a self-similar group G is contracting, the smallest finite set of G satisfying the above condition issaid to be the nucleus of G .The proposition below is included in the proof of [16, Theorem 7.3 (3)] Proposition 2.11.
Let G be a contracting countable self-similar group over X . Then we have µ (( X ω ) G -reg ) = 1 . It is not hard to show that the Grigorchuk group is contracting. We observe more examples ofcontracting self-similar groups in the next section.2.2. C ∗ -algebras arising from self-similar groups. We review the C ∗ -algebras introduced byNekrashevych constructed from self-similar groups in [17, 18]. Definition 2.12. ([17, Definition 3.1]) Define O G max to be the universal C ∗ -algebra generated by G (we assume that every relation in G is preserved) and { S x : x ∈ X } satisfying the following relationsfor any x ∈ X and g ∈ G :(2.2) g ∗ g = gg ∗ = 1 , (2.3) S ∗ x S x = 1 , N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 5 (2.4) X y ∈ X S y S ∗ y = 1 , (2.5) gS x = S g ( x ) g | x . The universal C ∗ -algebra O G max admits a nonzero representation. We recall the following represen-tations. Remark 2.13.
For each w ∈ X ω , h G, X i ( w ) denotes the h G, X i -orbit of w , i.e. h G, X i ( w ) = { f ( w ) : f ∈ h G, X i , w ∈ Dom f } . Let l ( h G, X i ( w )) be the set of l -functions on h G, X i ( w ). Write δ w for the characteristic functionof w ∈ h G, X i ( w ). We naturally identify each shift map T x with an isometry on l ( h G, X i ( w ))given by T x ( δ w ) := δ xw . Similarly, we identify each g ∈ G with a unitary on l ( h G, X i ( w )) givenby g ( δ w ) = δ g ( w ) . Thanks to the universality of O G max , we get a representation π w of O G max on l ( h G, X i ( w )) given by π w ( S x ) = T x , π w ( g ) = g for any x ∈ X, g ∈ G . In particular, O G max is nonzero.For u = x x · · · x n ∈ X n , we write S u := S x · · · S x n . Equations (2.3), (2.4) and Remark 2.13 implythat O G max contains the Cuntz algebra O | X | .We recall the following result due to V. V. Nekrashevych. Theorem 2.14. ([18, Theorem 3.3])
Let ρ be a unital representation of O G max on a nonzero Hilbertspace. Then for any w ∈ ( X ω ) S G -reg and a ∈ O G max , we have k π w ( a ) k ≤ k ρ ( a ) k . The above theorem implies that the C ∗ -algebra O G min := π w ( O G max ) does not depend on the choiceof w ∈ ( X ω ) S G -reg up to canonical isomorphisms.We identify h G, X i with { S u gS ∗ v : u, v ∈ X ∗ , g ∈ G } ⊂ O G max . For each w ∈ X ω , the restriction of π w to h G, X i is injective. Hence, we also write f for π w ( f ) ∈ O G min where w ∈ ( X ω ) S G -reg and f ∈ h G, X i . Let C h G, X i be the ∗ -subalgebra of O G max generated by G and { S x } x ∈ X . The self-similarity implies C h G, X i = span { S u gS ∗ v : u, v ∈ X ∗ , g ∈ G } . To understand O G min , we observe a Hilbert bimodule. Let A G min be the C ∗ -subalgebra of O G min generated by G . Define a subspace Φ of O G min byΦ := ( X x ∈ X S x a x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a x ∈ A G min ) . We regard Φ as a right A G min -module with the basis { S x } x ∈ X . Define an A G min -valued inner producton Φ by h X x ∈ X S x a x , X x ∈ X S x b x i := X x ∈ X a ∗ x b x where a x , b x ∈ A G min for any x ∈ X . Consider the left action of A G min on O G min arising from themultiplication. The left action provides a left A G min -module structure on Φ. Thus, we obtain a Hilbert A G min -bimodule. We also write Φ for this Hilbert A G min -bimodule.The following definition is almost the same as [17, Definition 6.1]. KEISUKE YOSHIDA
Definition 2.15. ([17, Definition 6.1]) Define O ′ G min to be the universal C ∗ -algebra generated by A G min and { S x : x ∈ X } satisfying equations (2.3), (2.4) and the following relation for any a ∈ A G min and x ∈ X :(2.6) aS x = X y ∈ X S y h S y , aS x i . Here the inner product is the A G min -valued one defined as above.The same argument as [17, Theorem 8.3] implies the simplicity of O ′ G min . Theorem 2.16. ([17, Theorem 8.3])
The universal C ∗ -algebra O ′ G min is unital, purely infinite andsimple. Let O (Φ) be the Cuntz–Pimsner algebra arising from the A G min -bimodule Φ. In fact, O ′ G min isisomorphic to O (Φ) and also isomorphic to O G min . Indeed, using the simplicity from Theorem 2.16,one can construct injective universal surjections from O ′ G min onto both O G min and O (Φ). Hence O ′ G min is isomorphic to O G min and O (Φ).We fix a definition and some notations related to the canonical gauge actions and KMS states. Definition 2.17.
Let D be a C ∗ -algebra. Fix a group action α : R y D . Assume that the map t α t ( c ) defined on R is norm-continuous for any c ∈ D . An element d ∈ D is said to be α -analytic if the map t α t ( d ) extends to an analytic map z α z ( d ) on C .In addition, fix a nonzero real number β . A state ϕ on D is said to be a ( β, α ) -KMS state if ϕ ( dc ) = ϕ ( cα iβ ( d ))for any c ∈ C and any α -analytic element d ∈ C .Under the assumption in Definition 2.17, we write D α := { d ∈ D : α t ( d ) = d for any t ∈ R } . Let Γ be the canonical gauge action of R on O G max given byΓ t ( g ) := g and Γ t ( S x ) := exp( it ) S x for g ∈ G , t ∈ R and x ∈ X . We also define the canonical gauge action Γ on O G min (we use the samesymbol as there is no confusion) byΓ t ( a ) := a and Γ t ( S x ) := exp( it ) S x for a ∈ A G min , t ∈ R and x ∈ X . Using the universalities, one can check the above gauge actions arewell-defined. We write h G, X i Γ := { S u gS ∗ v ∈ h G, X i : | u | = | v |} ⊂ h G, X i . Note that O Γ G max = span h G, X i Γ . Define a linear functional ψ on C h G, X i by ψ ( S u gS ∗ v ) := δ u,v | X | −| u | µ (( X ω ) g )where δ is the Kronecker’s delta. We have ψ ( f ) = µ (( X ω ) f ) for any f ∈ h G, X i by [23, Remark 2.9]. Lemma 2.18.
Let G be a self-similar group over X with µ (( X ω ) G -reg ) = 1 . Fix w ∈ ( X ω ) S G -reg .Then the map ψ ′ ( π w ( f )) := ψ ( f ) ; f ∈ h G, X i extends to a linear functional ψ ′ on π w ( C h G, X i ) . N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 7 Proof.
Take any η ∈ C h G, X i with π w ( η ) = 0. We prove ψ ( η ) = 0. Write η = X f ∈h G,X i γ f f where γ f = 0 for all but finitely many f ∈ h G, X i . One has ψ ( f ) = µ (( X ω ) f ) = Z X ω h δ w , f ( δ w ) i dµ ( w ) . for each f . Each inner product above is defined on l ( h G, X i ( w )). The assumption and Lemma 2.8imply µ (( X ω ) S G -reg ) = 1. Then a calculation shows ψ ( η ) = X f ∈h G,X i γ f Z X ω h δ w , f ( δ w ) i dµ ( w ) = X f ∈h G,X i γ f Z ( X ω ) S G -reg h δ w , f ( δ w ) i dµ ( w )= Z ( X ω ) S G -reg h δ w , ( π w ( η ))( δ w ) i dµ ( w ) = 0since π w ( η ) = 0 for any w ∈ ( X ω ) S G -reg . This proves the claim. (cid:3) In the case µ (( X ω ) G -reg ) = 1, the linear functionals ψ and ψ ′ extend to unique (log | X | , Γ)-KMSstates on O G max and O G min , respectively. See sections 6, 8 of [7] for the uniqueness of KMS states onToeplitz type C ∗ -algebra associated with self-similar groups. Theorem 2.19. ([7], [23, Theorems 3.11, 3.13])
Assume that G is a countable self-similar group over X with µ (( X ω ) G -reg ) = 1 . Then the following statements hold: (i) There exists a unique (log | X | , Γ) -KMS state ψ max on O G max . (ii) There exists a unique (log | X | , Γ) -KMS state ψ min on O G min . (iii) For any u, v ∈ X ∗ , g ∈ G , we have ψ max ( S u gS ∗ v ) = ψ min ( S u gS ∗ v ) = δ u,v µ (( X ω ) g ) . The uniqueness in Theorem 2.19 is not used in the next section.3.
Groupoid approaches and simplicity of O G max In this section, we observe the groupoids arising from self-simlar groups and their groupoid C ∗ -algebras. For some contracting self-similar groups, we discuss the simplicity of O G max through thegroupoid C ∗ -algebras.3.1. Groupoids arising from self-similar groups.
As the first part of this section, we recall thedefinitions and fix notations. A small category G whose any morphism has the inverse is said to bea groupoid . We identify a groupoid G with the set of morphisms and identify the objects with theidentity morphisms on them. We write the set of objects G (0) ⊂ G and refer to it as the unit space of G . Define two maps s, r from G onto G (0) by s ( γ ) = γ − γ and r ( γ ) = γγ − for γ ∈ G . The map s is called the source map and r is called the range map .A topological groupoid is a groupoid with a topology such that the multiplication operator andthe inverse operator are continuous. The source and range maps are continuous on any topologicalgroupoid. A topological groupoid is said to be ´etale (or r -discrete) if the unit space is locally compactwith respect to the relative topology of G and the source map is a local homeomorphism.A groupoid of germs of local homeomorphisms on a topological space is a good example of agroupoid. We observe the groupoids of germs arising from self-similar groups. Such groupoids havebeen introduced in [17]. KEISUKE YOSHIDA
Let ( h G, X i × X ω ) ′ be the subset of h G, X i × X ω given by( h G, X i × X ω ) ′ := { ( f, w ) ∈ h G, X i × X ω : w ∈ Dom f } We define an equivalence relation on ( h G, X i× X ω ) ′ as follows. Pairs ( f , w ) , ( f , w ) in ( h G, X i× X ω ) ′ are equivalent if w = w and f = f on a neighborhood of w = w . The quotient set is denoted by[ G, X ]. We write [ f, w ] for the equivalence class represented by ( f, w ).The set [
G, X ] forms a groupoid under the following operations. The multiplication is given by[ f , w ] · [ f , w ] := [ f f , w ]when w = f ( w ). The inverse is given by[ f, w ] − := [ f − , f ( w )] . We also equip [
G, X ] with the topology generated by sets of the form: U U,f = { [ f, w ] : w ∈ U } where f ∈ h G, X i and U is an open subset of Dom f . It is not hard to check that [ G, X ] is ´etale.The topological groupoid [
G, X ] might not be Hausdorff. For example, let X = { , } , G = h a, b, c, d i be the Grigorchuk group (see Example 2.4 for definition). Define1 ∞ := 11111 · · · ∈ X ω . We write e for the unit of G . Then [ e, ∞ ] = [ d, ∞ ] by definition but no open sets separate them. Tocheck this, note that d = e on 1 m X ω for any nonnegative integer m . For any open neighborhood U, V of 1 ∞ , there exists n ∈ N with 1 n X ω ⊂ U and 1 n X ω ⊂ V . We have U n X ω ,e ⊂ U n X ω ,e ⊂ U U,e and U n X ω ,d ⊂ U n X ω ,d ⊂ U V,d . In addition, we also have U n X ω ,e = U n X ω ,d = ∅ . Therefore, U U,e ∩ U
V,d = ∅ . This proves theclaim.Next we review the groupoid C ∗ -algebra. In this context, one often assumes that groupoids areHausdorff. In this paper, however, we do NOT assume the Hausdorffness to treat examples in subsec-tion 3.2. The definitions of groupoid C ∗ -algebras without Hausdorffness are introduced by A. Connesin [9].We again consider a (not necessarily Hausdorff) ´etale groupoid G . Let U ⊂ G be an open Hausdorffsubset. The set of continuous functions on U with compact support is denoted by C c ( U ). Let Funct( G )be the set of all functions on G . For η ∈ C c ( U ) , γ / ∈ U , we define η ( γ ) = 0. This gives an embedding of C c ( U ) into Funct( G ). We then define C ( G ) := span [ U C c ( U ) ⊂ Funct( G )where the union is taken over all Hausdorff open subsets U of G . Note that a function in C ( G ) mightnot be continuous on G .We define the multiplication and involution operators on C ( G ) by η ∗ ( γ ) := η ( γ − ) , ( η ∗ η )( γ ) := X γ γ = γ η ( γ ) η ( γ )for any η, η , η ∈ C ( G ). To introduce a C ∗ -norm on C ( G ), we fix the following notation. Let α ∈ G (0) and define G α := { γ ∈ G : s ( γ ) = α } . Consider the ∗ -representation λ α of C ( G ) on l ( G α ) given by( λ α ( η ) ζ )( γ ) = X γ γ = γ η ( γ ) ζ ( γ ) N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 9 for γ ∈ G α . It is not hard to check that k η k red = sup α ∈G (0) k λ α ( η ) k is a C ∗ -norm on C ( G ). The completion of C ( G ) with respect to k · k red is denoted by C ∗ r ( G ). TheC ∗ -algebra C ∗ r ( G ) is said to be the reduced groupoid C ∗ -algebra of G .For η ∈ C ( G ), we define the universal norm by k η k u := sup {k ρ ( η ) k : ρ is a ∗ -representation of C ( G ) on a Hilbert space } . This indeed defines a C ∗ -norm on C ( G ) (see [15, Lemma 1.2.3]). The completion of C ( G ) with respectto k · k u is denoted by C ∗ ( G ). The C ∗ -algebra C ∗ ( G ) is said to be the full groupoid C ∗ -algebra of G .We again consider the groupoid of germs arising from a self-similar group G over a finite set X .The universality of O G max provides a surjection from O G max onto C ∗ ([ G, X ]) given by h G, X i ∋ f U Dom f,f ∈ C ([ G, X ]) ⊂ C ∗ ([ G, X ])where 1 U Dom f,f is the characteristic function on U Dom f,f . It was shown by V. V. Nekrashevych thatthis surjection is in fact an isomorphism.
Theorem 3.1. ([17, Theorem 5.1], [12, Corollary 6.4])
The full groupoid C ∗ -algebra C ∗ ([ G, X ]) isisomorphic to O G max . Thanks to the theorem, one can identify f with 1 U Dom f,f . Moreover, by identifying C ∗ ([ G, X ])with O G max , we also consider the gauge action Γ defined in section 2 on C ∗ ([ G, X ]) and its quotient C ∗ r ([ G, X ]). Let [
G, X ] Γ := { [ S u gS ∗ v , w ] ∈ [ G, X ] : | u | = | v |} then the restriction of the isomorphism in Theorem 3.1 gives the following identification. Theorem 3.2. ([17, Theorem 5.3]) C ∗ ([ G, X ] Γ ) ≃ C ∗ ([ G, X ]) Γ ≃ O Γ G max . At the rest of this section, we discuss the simplicity of C ∗ ([ G, X ]) and C ∗ r ([ G, X ]) rather than O G max .If C ∗ ([ G, X ]) and C ∗ r ([ G, X ]) are not canonically isomorphic, then C ∗ ([ G, X ]) is not simple. Thus, weonly consider the case they are isomorphic.It is a well known fact that the full and reduced C ∗ -algebra of an amenable groupoid are canonicallyisomorphic. One can find this fact for Hausdorff groupoids in [1] or [19]. The same argument providesthe isomorphism in the non-Hausdorff case. We recall sufficient conditions of the amenability of [ G, X ]in [12] and [17].
Theorem 3.3. ([12, Corollary 10.18])
If a self-similar group G over X is amenable as a discrete group,then [ G, X ] is amenable. For the other sufficient condition, we recall the following definition.
Definition 3.4. ([17, Definition 2.3]) A self-similar group G over X is said to be self-replicating if forany x, y ∈ X, h ∈ G there exists g ∈ G with g ( x ) = y and g | x = h . Theorem 3.5. ([17, Theorem 5.6])
If a self-similar group G over X is contracting and self-replicating,then [ G, X ] is amenable. If [
G, X ] is amenable and Hausdorff, then C ∗ ([ G, X ]) is simple by standard arguments (see [17]).However, there exists a self-similar group G over X such that [ G, X ] is non-Hausdorff amenablegroupoid but the C ∗ ([ G, X ]) is not simple. We observe this later. In the next subsection, we considera class of self-similar groups whose groupoids are not Hausdorff. We discuss the simplicity of theirC ∗ -algebras. Multispinal groups.
From now on we consider a class of self-similar groups called multispinalgroups introduced in [20]. We first recall the construction.Let
A, B be finite groups and let B act freely on a finite set X . Write Aut( A ) and Hom( A, B ) forthe set of automorphisms of A and the set of homomorphisms from A to B , respectively. Consider amap Ψ from X to Aut( A ) ∪ Hom(
A, B ). Define A := Ψ( X ) ∩ Aut( A ) , B := Ψ( X ) ∩ Hom(
A, B ) . Set
B · A := B ∪ [ n ≥ { λ ◦ λ ◦ · · · ◦ λ n : λ ∈ B , λ i ∈ A for 2 ≤ i ≤ n } ⊂ Hom(
A, B ) . We assume
A 6 = ∅ , B 6 = ∅ and \ λ ∈B·A ker λ = { A } . Here 1 A is the unit of A . We define actions of A and B on X ω by a ( xw ) := x (Ψ( x )( a ))( w ) , b ( xw ) := b ( x ) w where a ∈ A, b ∈ B, x ∈ X, w ∈ X ω . The assumption implies that these two actions are faithful.Hence, we identify two finite groups A, B with their images in Homeo( X ω ), respectively. Let G be thesubgroup of Homeo( X ω ) generated by A and B . The group G is said to be a multispinal group over X . Let G = G ( A, B,
Ψ) denote the multispinal group arising from two finite groups A , B and a mapΨ. Put Y := Ψ − (Hom( A, B )) ⊂ X. Remark 3.6.
Let G = G ( A, B,
Ψ) be a multispinal group over X . By definition, the multispinalgroup G is a self-similar group. In fact, the multispinal group G is always contracting and the nucleuscoincides with A ∪ [ y ∈ Y Ψ( y )( A ) ⊂ A ∪ B. See [20, Proposition 7.1] for this fact. Assume that the action of B on X is transitive and [ y ∈ Y Ψ( y )( A ) = B. Then G is self-replicating (thus G is infinite) and the nucleus of G coincides with A ∪ B (See section7 of [20]). Therefore, [ G, X ] is amenable by Theorem 3.5.The Grigorchuk group is an example of a multispinal group.
Example 3.7.
Let A = Z × Z , B = X = Z . Consider the left translation action B y X . We defineΨ(0) ∈ Hom( Z × Z , Z ) and Ψ(1) ∈ Aut( Z × Z ) to beΨ(0)( x, y ) = y, Ψ(1)( x, y ) = ( y, x + y ) . Then the multispinal group G ( A, B,
Ψ) coincides with the Grigorchuk group generated by a, b, c, d (weuse the same symbols as in Example 2.4). To see this, it is sufficient to identify a with the generator1 ∈ Z and identify b, c, d with (0 , , (1 , , (1 , ∈ Z × Z , respectively.From now on, e is always the unit of a multispinal group G . Lemma 3.8.
Let G = G ( A, B, Ψ) be a multispinal group over X . Take w ∈ X ω , g, g ′ ∈ A ∪ B ⊂ G with g ( w ) = g ′ ( w ) . We write w = x x x · · · ; x , x , x , . . . ∈ X. Assume that w ∈ X ω uses an alphabet in Y and define m := min { i : x i ∈ Y } . Then g and g ′ coincide on the cylinder set w ( m ) X ω . In particular, we obtain [ g, w ] = [ g ′ , w ] . N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 11 Proof.
First we assume that g ∈ B \{ e } . Then by g ( w ) = g ′ ( w ), we have g = g ′ (otherwise the firstalphabet of g ( w ) and g ′ ( w ) do not coincide).Second, let g ∈ A . By the first step, we may assume g ′ ∈ A . Thus, it is sufficient to show that g and e coincide on w ( m ) X ω for g ∈ A with g ( w ) = w . By definition, x m ∈ Y and x i / ∈ Y for 1 ≤ i ≤ m − x m ) ◦ Ψ( x m − ) ◦ · · · ◦ Ψ( x ))( g ) ∈ B. Then, since g ( w ) = w , we obtain g ( w ( m ) ) = w ( m ) , g | w ( m ) = (Ψ( x m ) ◦ Ψ( x m − ) ◦ · · · ◦ Ψ( x ))( g ) = e. This shows the claim. (cid:3)
Lemma 3.9.
Let G = G ( A, B, Ψ) be a multispinal group over X . Then we have ( X \ Y ) ω ⊂ X ω \ ( X ω ) G -reg .More specifically, we have a ( w ) = a ′ ( w ) , [ a, w ] = [ a ′ , w ] for any w ∈ ( X \ Y ) ω , a, a ′ ∈ A with a = a ′ .Proof. It suffices to show that a ( w ) = w, [ a, w ] = [ e, w ]for w ∈ ( X \ Y ) ω , a ∈ A \{ e } . We write w = x x x · · · ; x , x , x , . . . ∈ X. By the definition of the action A y X ω , a does not change x . The assumption w ∈ ( X \ Y ) ω implies x ∈ X \ Y . Thus, Ψ( x )( a ) ∈ A . This shows that Ψ( x )( a ) does not change x ∈ X \ Y . Repeatingthis, we have a ( w ) = w .For each i , Ψ( x i ) is an automorphism on A . Therefore, we obtain a | w ( n ) = (Ψ( x n ) ◦ Ψ( x n − ) ◦ · · · ◦ Ψ( x ))( a ) = e for any n ∈ N . Thus [ a, w ] = [ e, w ]. (cid:3) Let G = G ( A, B,
Ψ) be a multispinal group over X . Recall that G is contracting [20, Proposition7.1]. Hence we have a faithful state ψ min on O G min by Proposition 2.11 and Theorem 2.19. We write π ψ min : O G min → B ( H ψ min ) , π ψ max : O G max → B ( H ψ max )for the GNS representations of ψ min and ψ max , respectively. Let w be an arbitrary strictly G -regularpoint. Note that we have ψ max ( η ) = ψ min ( π w ( η )) for any η ∈ O G max . Then we obtain a unitary U from H max onto H min given by U (ˆ η ) := \ π w ( η )for η ∈ O G max . Here ˆ η ∈ H max and \ π w ( η ) ∈ H min are equivalent classes represented by η and π w ( η ), respectively. The unitary U provides a unitarily equivalence between ( π ψ min ◦ π w , H ψ min ) and( π ψ max , H ψ max ). In addition, π ψ min is faithful since O G min is simple by Theorem 2.16. Thus, we have k π w ( η ) k = k π ψ min ( π w ( η )) k = k π ψ max ( η ) k for any η ∈ O G max .Write C ∗ ( A ) for the full group C ∗ -algebra of A . Since A is finite, the canonical inclusion A → O G max extends to an embedding C ∗ ( A ) → span( A ) ⊂ O G max . From now on, we identify C ∗ ( A ) with the finite dimensional C ∗ -subalgebra span( A ) of O G max . Notethat the restriction of ψ max to O Γ G max induces a tracial state. Lemma 3.10.
Let G = G ( A, B, Ψ) be a multispinal group over X . Assume that the restriction of ψ max to C ∗ ( A ) is faithful. Then there exists an orthonormal system { a } a ∈ A in H ψ max with π ψ max ( a ) a ′ = aa ′ for any a, a ′ ∈ A .Proof. Let τ be the canonical tracial state on C ∗ ( A ): τ ( a ) = δ a,e for any a ∈ A . Here δ is the Kronecker delta. Recall that τ is faithful. In addition, the restriction of ψ max to C ∗ ( A ) is a faithful tracial state on C ∗ ( A ) by assumption. Note that C ∗ ( A ) is finite dimensional.Therefore, the non-commutative Radon–Nikodym theorem provides a positive invertible element h in C ∗ ( A ) with τ ( k ) = ψ max ( hk )for any k ∈ C ∗ ( A ) (see [22, Corollary 3.3.6]). Define a := ah for each a ∈ A . Note that the restriction of the canonical map O G max → H ψ max to C ∗ ( A ) is injectiveby assumption. We regard each a as a unit vector in H ψ max via the canonical map. Clearly, π ψ max ( a ) a ′ = aa ′ for any a, a ′ ∈ A . Considering the inner product h· , ·i on H ψ max , we have h a, e i = ψ max ( ah ) = ψ max ( ha ) = τ ( a ) = δ a,e for any a ∈ A . This shows the claim. (cid:3) We fix notations for the proof of the main theorem of this section. We observe the reduced groupoidC ∗ -algebra C ∗ r ([ G, X ] Γ ). Consider the map w [ e, w ]from X ω to [ G, X ] Γ . It is not hard to show that the map provides a homeomorphism from X ω ontothe unit space ([ G, X ] Γ ) (0) . For [ e, w ] ∈ ([ G, X ] Γ ) (0) , we write w instead of [ e, w ]. Then the C ∗ -normon C ∗ r ([ G, X ] Γ ) is given by sup w ∈ X ω k λ Γ w ( · ) k . Let C h G, X i Γ be the ∗ -subalgebra of O G max generated by h G, X i Γ . Note that for each w ∈ X ω , wehave ([ G, X ] Γ ) w = { [ f, w ] : f ∈ h G, X i Γ , w ∈ Dom f } . Let δ [ f,w ] ∈ l (([ G, X ] Γ ) w ) be the characteristic function of [ f, w ] ∈ ([ G, X ] Γ ) w . Then we have λ Γ w ( f ) δ [ f ,w ] = ( δ [ f f ,w ] if f ( w ) ∈ Dom f f ∈ h G, X i Γ and [ f , w ] ∈ [ G, X ] Γ . Remark 3.11.
Let G be a self-similar group over X . For any S u gS ∗ v ∈ h G, X i Γ with | u | = | v | < n ,we have(3.1) g = g X u ′ ∈ X n −| u | S u ′ S ∗ u ′ = X u ′ ∈ X n −| u | S g ( u ′ ) g | u ′ S ∗ u ′ . Consider any finite sum(3.2) η = X g,u,v, | u | = | v | α gu,v S u gS ∗ v ∈ C h G, X i Γ . Using (3.1), we may assume that the sum is taken over u, v ∈ X n for some fixed n ∈ N . N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 13 Next, we further assume that G is a contracting self-similar group with the nucleus N . Then forany finite subset F ′ ⊂ G , there exists a natural number n ∈ N satisfying g ′ | u ′ ∈ N for any g ′ ∈ F ′ and u ′ ∈ X ∗ with | u ′ | > n . Thus, for any g ′ ∈ F ′ , n ′ > n , we have g ′ = g ′ X u ′ ∈ X n ′ S u ′ S ∗ u ′ = X u ′ ∈ X n ′ S g ′ ( u ′ ) g ′ | u ′ S ∗ u ′ . This shows that one can further assume that the sum in (3.2) is taken over g ∈ N , u, v ∈ X n for somefixed (larger) n ∈ N .For finite words u, v ∈ X ∗ , uv is also a finite word. For short, we temporary write u − ( uv ) := v. Let w ∈ X ω and n, m be natural numbers with n > m . We write w − ( m )+( n ) := ( w ( m ) ) − w ( n ) ∈ X n − m . Remark 3.12.
Let G be a self-similar group over X . Take any w ∈ X ω . For any [ f, w ] ∈ ([ G, X ] Γ ) w ,there exist u ∈ X ∗ , g ∈ G and m ∈ N with | u | = m, f = S u gS ∗ w ( m ) . For any natural number n with n > m , we have(3.3) [ S u gS ∗ w ( m ) , w ] = [ S u gS ∗ w ( m ) S w ( n ) S ∗ w ( n ) , w ] = [ S u S g ( w − ( m )+( n ) ) g | w − ( m )+( n ) S ∗ w ( n ) , w ] . Assume that G is a contracting self-similar group with the nucleus N . Take finitely supportedfunction ξ ∈ l (([ G, X ] Γ ) w ). Then, by the equation (3.3) and similar arguments to Remark 3.11, onecan write(3.4) ξ = X u ′ ∈ X n ′ ,g ′ ∈N β g ′ u ′ δ [ S u ′ g ′ S ∗ w ( n ′ ) ,w ] for some fixed natural number n ′ and complex numbers β g ′ u ′ . Note that one can replace n ′ to bearbitrarily large.In the following theorem, we assume that [ G, X ] is amenable. See Theorem 3.3 and Remark 3.6 forthe sufficient condition of the amenability of [
G, X ]. Theorem 3.13.
Let G = G ( A, B, Ψ) be a multispinal group over X . Assume that [ G, X ] is amenable.Then the following conditions are equivalent. (i) The restriction of ψ max to C ∗ ( A ) is faithful. (ii) The | A | × | A | matrix [ ψ ( a − a )] a ,a ∈ A is invertible. (iii) The universal C ∗ -algebra O G max is simple.Proof. The equivalence between (i) and (ii) is given by standard arguments.If we assume (iii), then π ψ max is faithful by the simplicity of O G max . Recall that ψ max is a KMSstate on O G max . Therefore, ψ max is faithful by [5, Corollary 5.3.9]. This proves (iii) ⇒ (i).We show (i) ⇒ (iii). Combining the amenability of [ G, X ] and Theorem 3.1, we have the identifica-tions O G max ≃ C ∗ ([ G, X ]) ≃ C ∗ r ([ G, X ]) . We prove O G min ≃ C ∗ r ([ G, X ]). Recall that O G min is a simple quotient of C ∗ r ([ G, X ]) (Theorems 2.14,2.16). It is sufficient to show the injectivity of this quotient map. Since the gauge action Γ is periodic,we only check the injectivity on the gauge invariant subalgebra C ∗ r ([ G, X ]) Γ ≃ C ∗ r ([ G, X ] Γ ) (see [6,Proposition 4.5.1]). We do this by the norm estimate.Take any w ∈ X ω and any nonzero element η ∈ C h G, X i Γ . The goal of this proof is to show k λ Γ w ( η ) k ≤ k π ψ max ( η ) k = k π w ( η ) k . Here w is a strictly G -regular point which we choose later. Recall that we have k π ψ max ( η ) k = k π w ′ ( η ) k for any w ′ ∈ ( X ω ) S G -reg . From now on, we write ψ for ψ max . By the definition of operator norm, for any ε > ξ ∈ l ([ G, X ] Γ w ) with k λ Γ w ( η ) ξ k ≥ (1 − ε ) k λ Γ w ( η ) kk ξ k . Write η = X u,v,g, | u | = | v | α gu,v S u gS ∗ v , ξ = X u ′ ,g ′ β g ′ u ′ δ [ S u ′ g ′ S ∗ w ( | u ′| ) ,w ] . Here δ [ S u ′ g ′ S ∗ w ( | u ′| ) ,w ] is the characteristic function on [ S u ′ g ′ S ∗ w ( | u ′| ) , w ]. Note that G is contracting [20,Proposition 7.1]. We write N for the nucleus of G , namely N := A ∪ [ y ∈ Y Ψ( y )( A ) ⊂ A ∪ B. By Remarks 3.11, 3.12, we may assume that the finite sums are taken over g, g ′ ∈ N and u, v, u ′ ∈ X n for some fixed n ∈ N . We write w = x x x · · · ; x , x , x , . . . ∈ X. We divide the proof into two cases. For the first case, assume that w uses infinitely many alphabetsin Y . Then we have a natural number n ′ with n ′ > n such that there exists i ∈ N with n < i < n ′ and x i ∈ Y . Note that we have bS x = S b ( x ) , aS yx = S y S (Ψ( y )( a ))( x ) for any x ∈ X, y ∈ Y, a ∈ A, b ∈ B . This shows[ S u ′ g ′ S ∗ w ( n ) , w ] = [ S u ′ g ′ S ∗ w ( n ) S w ( n ′ ) S ∗ w ( n ′ ) , w ]= [ S u ′ S g ′ ( w − ( n )+( n ′ ) ) g ′ | w − ( n )+( n ′ ) S ∗ w ( n ′ ) , w ]= [ S u ′ S g ′ ( w − ( n )+( n ′ ) ) S ∗ w ( n ′ ) , w ]for any u ′ ∈ X n , g ′ ∈ N ⊂ A ∪ B . Now replace n by n ′ and rewrite η = X g ∈N ,u,v ∈ X n α gu,v S u gS ∗ v , ξ = X u ′ ∈ X n β u ′ δ [ S u ′ S ∗ w ( n ) ,w ] . Then, we compute λ Γ w ( η ) ξ = X g ∈N ,u,v,u ′ ∈ X n α gu,v β u ′ λ Γ w ( S u gS ∗ v ) δ [ S u ′ S ∗ w ( n ) ,w ] = X g,u,v α gu,v β v δ [ S u gS ∗ w ( n ) ,w ] . Let m be the smallest natural number with x n + m ∈ Y . Using the density of ( X ω ) G -reg in X ω , wechoose a G -regular point w ∈ w ( n + m ) X ω . Define a vector ˜ ξ ∈ l ( h G, X i ( w )) by˜ ξ := X u ′ ∈ X n β u ′ δ ( S u ′ S ∗ w ( n ) ( w )) . A calculation shows π w ( η ) ˜ ξ = X g ∈N ,u,v,u ′ ∈ X n α gu,v β u ′ π w ( S u gS ∗ v ) δ ( S u ′ S ∗ w ( n ) ( w )) = X g,u,v α gu,v β v δ ( S u gS ∗ w ( n ) ( w )) . Note that n δ [ S u ′ S ∗ w ( n ) ,w ] o u ′ ∈ X n and n δ ( S u ′ S ∗ w ( n ) ( w ) o u ′ ∈ X n are orthonormal systems in l (([ G, X ] Γ ) w )and l ( h G, X i ( w )), respectively. Thus, we have k ξ k = k ˜ ξ k . To compare the norms k λ Γ w ( η ) ξ k and k π w ( η ) ˜ ξ k , we prove the following claim. Claim.
Take any u , u ∈ X n , g , g ∈ A ∪ B . Then the following conditions are equivalent. (a) [ S u g S ∗ w ( n ) , w ] = [ S u g S ∗ w ( n ) , w ] . (b) S u g S ∗ w ( n ) ( w ) = S u g S ∗ w ( n ) ( w ) . N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 15 Proof of Claim.
First, we assume (a). Then we have u = u and g S ∗ w ( n ) ( w ) = g S ∗ w ( n ) ( w ). Hence,Lemma 3.8 implies that g S ∗ w ( n ) and g S ∗ w ( n ) coincide on w ( n + m ) X ω from the choice of m . Since w ∈ w ( n + m ) X ω , we obtain (b).Conversely, we assume (b). Then u = u and g S ∗ w ( n ) ( w ) = g S ∗ w ( n ) ( w ). Hence, Lemma 3.8shows that g S ∗ w ( n ) and g S ∗ w ( n ) coincide on w ( n + m )0 X ω = w ( n + m ) X ω . This implies (a). (cid:3) The claim implies k λ Γ w ( η ) ξ k = k π w ( η ) ˜ ξ k . Thus, k π w ( η ) kk ˜ ξ k ≥ k π w ( η ) ˜ ξ k = k λ Γ w ( η ) ξ k ≥ (1 − ε ) k λ Γ w ( η ) kk ξ k = (1 − ε ) k λ Γ w ( η ) kk ˜ ξ k . Since ε is arbitrary, we have k π w ( η ) k ≥ k λ Γ w ( η ) k .For the second case, we assume that w uses at most finitely many alphabets in Y . Choose a naturalnumber n ′ with S ∗ w ( n ′ ) ( w ) ∈ ( X \ Y ) ω and n ′ > n . Then we have[ S u ′ g ′ S ∗ w ( n ) , w ] = [ S u ′ g ′ S ∗ w ( n ) S w ( n ′ +2) S ∗ w ( n ′ +2) , w ]= [ S u ′ S g ′ ( w − ( n )+( n ′ +2) ) g ′ | w − ( n )+( n ′ +2) S ∗ w ( n ′ +2) , w ]and g ′ ( x n ′ +2 ) ∈ X \ Y, g ′ | w − ( n )+( n ′ +2) ∈ A for any g ′ ∈ N ⊂ A ∪ B . For each l ∈ N , define Z l := { u = z z · · · z l ∈ X l : z l ∈ X \ Y } . Note that for any g ∈ A ∪ B, u ∈ Z l , we have g | u ∈ A . Thus, one can rewrite ξ = X u ′ ∈ Z n ′ +2 ,g ′ ∈ A β g ′ u ′ δ [ S u ′ g ′ S ∗ w ( n ′ +2) ,w ] . Take an orthonormal system { g } g ∈ A given in Lemma 3.10. Define a vector ˜ ξ ∈ H ψ by˜ ξ := X u ′ ∈ Z n ′ +2 ,g ′ ∈ A β g ′ u ′ π ψ ( S u ′ ) g ′ . Take any g ′ , g ′ ∈ A, u ′ , u ′ ∈ Z n ′ +2 . By Lemma 3.9, [ S u ′ g ′ S ∗ w ( n ′ +2) , w ] = [ S u ′ g ′ S ∗ w ( n ′ +2) , w ] if and onlyif S u ′ = S u ′ , g ′ = g ′ . Thus, k ξ k = k ˜ ξ k .For each u ′ = z ′ z ′ · · · z ′ n ′ +2 ∈ Z n ′ +2 we define u ′ ( n ) := z ′ z ′ · · · z ′ n ∈ X n . Then for any g ∈ N , g ′ ∈ A, u, v ∈ X n , u ′ ∈ Z n ′ +2 with u ′ ( n ) = v , we have λ Γ w ( S u gS ∗ v ) δ [ S u ′ g ′ S ∗ w ( n ′ +2) ,w ] = X v ′ ∈ X n ′ +2 − n λ Γ w ( S u gS v ′ S ∗ v ′ S ∗ v ) δ [ S u ′ g ′ S ∗ w ( n ′ +2) ,w ] = δ [ S ug ( v − u ′ ) g | v − u ′ g ′ S ∗ w ( n ′ +2) ,w ] . Note that g | v − u ′ ∈ A since v − u ′ ∈ Z n ′ +2 − n . Similarly, we obtain π ψ ( S u gS ∗ v ) (cid:0) π ψ ( S u ′ ) g ′ (cid:1) = π ψ ( S ug ( v − u ′ ) ) g | v − u ′ g ′ . In addition, for any g ∈ N , g ′ ∈ A, u, v ∈ X n , u ′ ∈ Z n ′ +2 with u ′ ( n ) = v , we have λ Γ w ( S u gS ∗ v ) δ [ S u ′ g ′ S ∗ w ( n ′ +2) ,w ] = 0 , π ψ ( S u gS ∗ v ) (cid:0) π ψ ( S u ′ ) g ′ (cid:1) = 0 . Hence, we get λ Γ w ( η ) ξ = X u,v,u ′ ,g,g ′ α gu,v β g ′ u ′ δ [ S ug ( v − u ′ ) g | v − u ′ g ′ S ∗ w ( n ′ +2) ,w ] . and π ψ ( η ) ˜ ξ = X u,v,u ′ ,g,g ′ α gu,v β g ′ u ′ π ψ ( S ug ( v − u ′ ) ) g | v − u ′ g ′ . Here the sums are taken over g ∈ N , g ′ ∈ A, u, v ∈ X n , u ′ ∈ Z n ′ +2 with u ′ ( n ) = v . We claim k λ Γ w ( η ) ξ k = k π ψ ( η ) ˜ ξ k . Define T := { S v ′ h : h ∈ A, v ′ ∈ Z n ′ +2 } ⊂ h G, X i . Take any S v ′ h , S v ′ h ∈ T . By Lemma 3.9, [ S v ′ h S ∗ w ( n ′ +2) , w ] = [ S v ′ h S ∗ w ( n ′ +2) , w ] if and only if S v ′ = S v ′ , h = h . Hence, [ S v ′ h S ∗ w ( n ′ +2) , w ] = [ S v ′ h S ∗ w ( n ′ +2) , w ] if and only if π ψ ( S v ′ ) h = π ψ ( S v ′ ) h .Now, we rewrite(3.5) λ Γ w ( η ) ξ = X S v ′ h ∈ T X S ug ( v − u ′ ) g | v − u ′ g ′ = S v ′ h α gu,v β g ′ u ′ δ [ S v ′ hS ∗ w ( n ′ +2) ,w ] , (3.6) π ψ ( η ) ˜ ξ = X S v ′ h ∈ T X S ug ( v − u ′ ) g | v − u ′ g ′ = S v ′ h α gu,v β g ′ u ′ π ψ ( S v ′ ) h. The equations (3.5, 3.6) shows the claim. Consequently, we have k π ψ ( η ) k ≥ k λ Γ w ( η ) k from the sameargument as the first case. Thus we finished the proof. (cid:3) Corollary 3.14.
Let G = G ( A, B, Ψ) be a multispinal group over X . Assume that [ G, X ] is amenable.If one of the equivalent conditions of Theorem 3.13 holds, then the universal C ∗ -algebra O G max is aKirchberg algebra.Proof. Since the assumption implies that O G max is simple, we have O G max ≃ O G min . Thus, O G max is purely infinite simple by Theorem 2.16. By assumption, [ G, X ] is amenable. Thenas C ∗ ([ G, X ]) is nuclear, so is O G max . Trivially, O G max is separable and unital. Hence, O G max is aKirchberg algebra. (cid:3) For the other corollary, we fix notations. Consider a multispinal group G = G ( A, B,
Ψ). We write C ∗ ( B ) for the full group C ∗ -algebra of the finite group B . In the same way as C ∗ ( A ), we identify C ∗ ( B )with a finite dimensional C ∗ -subalgebra of O G max . For each λ ∈ Hom(
A, B ), define a ∗ -homomorphism˜ λ : C ∗ ( A ) → C ∗ ( B ) to be ˜ λ X a ∈ A α a a ! := X a ∈ A α a λ ( a ) . Here each α a is a complex number. We recall a spacial case of [20, Theorem 7.5]. Theorem 3.15. ([20, Theorem 7.5])
Let G = G ( A, B, Ψ) be a multispinal group over X . Then C h G, X i is simple if and only if \ λ ∈B·A ker ˜ λ = { } . Combining Theorems 3.13, 3.15, we have the following corollary.
Corollary 3.16.
Let G = G ( A, B, Ψ) be a multispinal group over X . Assume that [ G, X ] is amenable.Then O G max is simple if and only if C h G, X i is simple.Proof. First, we show the simplicity of C h G, X i from the simplicity of O G max . In sections 2, 4 of [20],it was shown that C h G, X i is isomorphic to the Steinberg C -algebra associated to [ G, X ]. Thus, thesimplicity of O G max ≃ C ∗ ([ G, X ]) implies the simplicity of C h G, X i by [8, Corollary 4.12].Next, we prove the simplicity of O G max from the simplicity of C h G, X i . We check the condition (i)of Theorem 3.13. Take any nonzero positive element η ∈ C ∗ ( A ). Write η = X a ∈ A α a a. N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 17 Since η = 0, there exists a homomorphism λ ∈ B · A with ˜ λ ( η ) = 0 by Theorem 3.15. In the case λ / ∈ B , write λ = Ψ( x n ) ◦ Ψ( x n − ) ◦ · · · ◦ Ψ( x ) , u = x x · · · x n . Here n ≥ , x n ∈ Y and x i ∈ X \ Y for any 1 ≤ i ≤ n −
1. In the case λ ∈ B , we write λ = Ψ( x ) and u = x ∈ Y . For each a ∈ A , we have aS u = aS x S x · · · S x n = S x Ψ( x )( a ) S x · · · S x n = · · · = S u λ ( a ) . This shows that(3.7) ηS u S ∗ u = X a ∈ A α a a ! S u S ∗ u = S u X a ∈ A α a λ ( a ) ! S ∗ u = S u ˜ λ ( η ) S ∗ u . Note that ˜ λ ( η ) is a nonzero positive element in C ∗ ( B ). We write(˜ λ ( η )) = X b ∈ B β b b. Here β b = 0 for some b ∈ B . Since ψ max ( b ) = µ (( X ω ) b ) = 0 for any b = e , we have(3.8) ψ max (˜ λ ( η )) = ψ max X b ∈ B β b b ! ∗ X b ∈ B β b b !! = X b ∈ B | β b | > . Recall that the restriction of ψ max to O Γ G max is tracial. Then a calculation shows(3.9) ψ max ( ηS u S ∗ u ) = ψ max ( η S u S ∗ u η ) ≤ ψ max ( η η ) = ψ max ( η ) . Combining (3.7), (3.8), (3.9) and Theorem 2.19, we obtain ψ max ( η ) ≥ ψ max ( ηS u S ∗ u ) = ψ max ( S u ˜ λ ( η ) S ∗ u ) = | X | −| u | ψ max (˜ λ ( η )) > . This proves the claim. (cid:3)
Using Theorem 3.13, we show the simplicity of O G max in the case that G is the Grigorchuk group.The simplicity of O G max of the Grigorchuk group has been proved in [8, Theorem 5.22]. The proof in[8] is based on its main theorem. The main theorem provides a relevance between supports of functionson a non-Hausdorff groupoid and the simplicity of its reduced groupoid C ∗ -algebra (see [8, Theorem4.10]). We observe a different proof in the following example. Example 3.17.
We regard the Grigorchuk group as a multispinal group (see Example 3.7). In thisexample, we use the same symbols as in Example 2.4 for generators. In other words, we write Z × Z = { e, b, c, d } , Z = { e, a } . Note that we have b = c = d = e, bc = cb = d, bd = db = c, cd = dc = b. We determine the values ψ ( b ) , ψ ( c ) , ψ ( d ) to check the condition (ii) in Theorem 3.13. This is done ifone solves the following simultaneous equation given by the self-similarity: ψ ( b ) = ψ ( a ) + ψ ( c )2 = ψ ( c )2 ,ψ ( c ) = ψ ( a ) + ψ ( d )2 = ψ ( d )2 ,ψ ( d ) = ψ ( e ) + ψ ( b )2 = 1 + ψ ( b )2 . As the solution of these equations, we have ψ ( b ) = 17 , ψ ( c ) = 27 , ψ ( d ) = 47 . Thus, ψ ( e ) ψ ( b ) ψ ( c ) ψ ( d ) ψ ( b ) ψ ( e ) ψ ( d ) ψ ( c ) ψ ( c ) ψ ( d ) ψ ( e ) ψ ( b ) ψ ( d ) ψ ( c ) ψ ( b ) ψ ( e ) = 17 . Compute the determinant of the matrixdet = 896 . By Theorem 3.13, the universal C ∗ -algebra O G max is simple.In the next example, we observe a non-simple case. Combining [8, Corollary 4.12] and [20, Example7.6], one can get the same result. Example 3.18.
Let A = Z × Z , B = Z and X = { , } . Consider the left translation action B y X . We define Ψ(0) ∈ Hom( Z × Z , Z ) and Ψ(1) ∈ Aut( Z × Z ) to beΨ(0)( x, y ) := y, Ψ(1)( x, y ) := ( y, x ) . Respecting the Grigorchuk group, we write b := (0 , , c := (1 , , d := (1 ,
0) and let a denote thegenerator of Z . Then one obtains a (0 w ) = 1 w, a (1 w ) = 0 w,b (0 w ) = 0 a ( w ) , b (1 w ) = 1 d ( w ) ,c (0 w ) = 0 a ( w ) , c (1 w ) = 1 c ( w ) ,d (0 w ) = 0 w, d (1 w ) = 1 b ( w )for any w ∈ X ω . These equations give ψ ( b ) = ψ ( a ) + ψ ( d )2 = ψ ( d )2 ,ψ ( c ) = ψ ( a ) + ψ ( c )2 = ψ ( c )2 ,ψ ( d ) = ψ ( e ) + ψ ( b )2 = 1 + ψ ( b )2 . Then we get ψ ( b ) = 13 , ψ ( c ) = 0 , ψ ( d ) = 23 . Thus, we havedet ψ ( e ) ψ ( b ) ψ ( c ) ψ ( d ) ψ ( b ) ψ ( e ) ψ ( d ) ψ ( c ) ψ ( c ) ψ ( d ) ψ ( e ) ψ ( b ) ψ ( d ) ψ ( c ) ψ ( b ) ψ ( e ) = det = 0 . Theorem 3.13 implies that the universal C ∗ -algebra O G max is not simple in this case. Example 3.19.
Next we observe the case A = Z × Z , B = X = Z . The action of B on X is the left translation action. We regard the elements in A as Z -valued row vectors. We defineΨ(0) , Ψ(1) ∈ Aut( Z × Z ) and Ψ(2) ∈ Hom( Z × Z , Z ) to beΨ(0) = (cid:18) (cid:19) , Ψ(1) = (cid:18) (cid:19) , Ψ(2) = (cid:0) (cid:1) . Let a = (cid:18) (cid:19) , a = (cid:18) (cid:19) , a = (cid:18) (cid:19) , a = (cid:18) (cid:19) , a = (cid:18) (cid:19) . N THE SIMPLICITY OF C ∗ -ALGEBRAS ASSOCIATED TO MULTISPINAL GROUPS 19 By the self-similarity, we have ψ ( a ) = ψ ( a ) + ψ ( a ) + 13 ,ψ ( a ) = ψ ( a ) + ψ ( a )3 ,ψ ( a ) = 2 ψ ( a )3 ,ψ ( a ) = ψ ( a ) + ψ ( a )3 . From these equations, we obtain ψ ( a ) = ψ ( a − ) = 47 ,ψ ( a ) = ψ ( a − ) = 114 ,ψ ( a ) = ψ ( a − ) = 17 ,ψ ( a ) = ψ ( a − ) = 314 . Hence we obtain the matrix[ ψ ( a − i a j )] ≤ i,j ≤ = 114
14 1 8 2 3 1 8 2 31 14 3 8 2 1 2 3 88 3 14 1 1 2 8 3 22 8 1 14 1 3 3 2 83 2 1 1 14 8 2 8 31 1 2 3 8 14 3 8 28 2 8 3 2 3 14 1 12 3 3 2 8 8 1 14 13 8 2 8 3 2 1 1 14 . Here a := a − , a := a − , a := a − , a := a − . The MDETERM function of Excel provides the resultof the calculation as follows:det
14 1 8 2 3 1 8 2 31 14 3 8 2 1 2 3 88 3 14 1 1 2 8 3 22 8 1 14 1 3 3 2 83 2 1 1 14 8 2 8 31 1 2 3 8 14 3 8 28 2 8 3 2 3 14 1 12 3 3 2 8 8 1 14 13 8 2 8 3 2 1 1 14 = 634894848 . As a consequence, we get the simplicity of O G max in this case. Acknowledgements.
The author appreciates his supervisors, Yuhei Suzuki and Reiji Tomatsu, forfruitful discussions and their constant encouragements.
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