Boundary Action On Simple Reduced Group C*-Algebras

A connection between boundary actions, ideal structure of reduced crossed products and C*-simple group is imminent.We investigate the stability properties for discrete group pioneered by powers and show that the non-abelian free group on two generators is C*-simple.Kalantar and Kennedy [32, Theorem 6.2] is now extended. Some examples are given using characterization of C*-simplicity obtained by Kalantar, Kennedy, Breuillard, and Ozawa [10, Theorem 3.1]


INTRODUCTION
Let G be a discrete group. Let the group algebra l1(G) equipped with the following product and involution From now on, we write the notation x=∑ g∈G xgδg for a functionx∈l1(G,A), where xg=x(g) for g∈G. We next equip l1(G,A) with a product and involution by defining (xy)(s)=∑ g∈G x(g)(gy(g−1s)),x¿(s)=sx(s−1) So that l1(G,A) becomes a Banach *-algebra in the 1-norm. We identify A with the image of A under the *-homomorphism a↦aδ1. Obviously, the subset cc(G,A) of finitely supported functorsG→A is a dense *-subalgebra of l1(G,A) and that an approximate identity (ei) in A yields an approximate identity (eiδ1)∈l1(G,A). It is well known that a covariant representation of the c*-dynamical system (A,G,α) is a triple (π,u,H), where H is a Hilbert space, π∶A→B(H) is a non-degenerate representation and u∶G→U(H) is a unitary representation such that π(ga)=ugaug ¿ for g∈G and a∈A. We often suppress the Hilbert space H from the notationif is clear from the context. The associated integral form of covariant representation (π,u) is the map π×u∶l1(G,A)→B(H) define by (π×u)(x)=∑ g∈G π(xg)ug,x∈l1(G,A) The full cross product of (A,G,α), denoted by A⋊αG=A⋊G is the completion of l1(G,A) or cc(G,A) with respect to the norm ‖x‖u =‖(π×u)(x)‖ ,x∈l1(G,A) The supremum taken over all (cyclic) covariant representations (π,u,H) of (A,G,α). To define the reduced crossed product, we assume thatA⊆B(H) is faithfully represented and define and we define a faithful representationπ∶A→B(H⊗l2(G)) and a unitary representation λ∶G→B(H⊗l2(G)) by π(a)(ξ⊗δs)=(s−1a)ξ⊗δs,λg(ξ⊗δs)=ξ⊗δgs,a∈A,ξ∈H,gs∈G It is verifiable,(π,λ,H⊗l2(G)) is a covariant representation of (A,G,α) call a regular representation of the c*-dynamical system. Again, is actually an amplification of the left regular representation of G on l2(G). The associated form π×λ∶l1(G)→B(H⊗l2(G)) is faithful, and the reduced crossed product A⋊α,rG=A⋊rG is the completion of l1(G,A) or cc(G,A) in the reduced norm It turns that a property of φ is inherited by ~ φ. It is easy to show that this include faithfulness, surjectivity and being a *-homomorphism.

Definition 1.2
For an action of a discrete group G on a topological free space X, define Xg={x∈X|gx=x},g∈G, we say that the action of G on X is topologically free if Xg has empty interior for all g∈G∖{1}.
Moreover, if B⊆A is a c*-algebras and φ:A→B is a c.c.p map that restricts to the identity map on ℬ, then φ is in fact is conditional of A onto ℬ.

Lemma 1.4(Archbold and Spielberge [1, Theorem 1]):
LetX be a compact G-space on which the action of G is topologically free. If I⊆C(X)⋊rG is closed ideal such that I∩C(X)={0}, then I={0}. For purpose of clarity, we omit the proof of Theorem 1.5. ). If f:X→X is homeomorphism, then the fixed point set of f is clopen. In particular, a group action on X is topologically free if and only if it is free. Boundary action are intimately connected with several commutative C*-algebras that are of interest in the study of c*-simple groups (i.e. groups with simple reduced group c*-algebras) can be found in the literatures [16, 18, 24, 26, 31, 32, 40, and 42]. The concept of boundary action was originally introduced by Furstenberg [24]. The main idea is to describe to what degree a fixed group of homeomorphism of space (i.e., a fixed non-trivial translation of R to any bounded subset) can map any or at least some points in the boundary of R∈R¿, namely (−∞,+∞) in space. It is clear that any non-trivial translation of R with positive derivative move any point in R∪{∞} closer to +∞, and that{±∞} are the only fixed point. The study of boundary actions and ideal structure of reduced crossed products have recently be linked to the study of c*-simple group (see. [16]). Furthermore, a discrete group can only be c*-simple when the c*-algebra associated to its regular representation is simple. This property for discrete group pioneered by powers will be one of the focuses of this paper. In particular, Our motivations are the advances in [19, 20, and 21] which were later elaborated in [28].It is our purpose in this paper to extend the Theorem of Kalanter and Kennedy [32, Theorem 6.2] that characterizes c*-simplicity in terms of boundary actions to the equivalence of topological freeness due to Kalanter, Kennedy, Breuillard and Ozawa [10, Theorem 3.1] and then generalized some of their results.We remark that other characterization of c*-simplicity have been obtained since the result of Kennedy and Kalantar. A few of which we now review; I. Simplicity of reduced crossed products. Breuillard, Kalantar, Kennedy and Ozawa proved that c*simple discrete groups have the property that a reduced crossed product A⋊rG of a unital G-c*-algebra by G is simple if and only if A is G-simple, which means that A has no non-trivial Ginvariant closed ideals [10, Theorem 7.1]. This settled in affirmative a question of de la Harpe and Skandalis [21]. We will generalize this result in section 3. II. An averaging property. Haagerup [29] and Kennedy [36] independently proved that a discrete group G is c*-simple if and only if for all t1,t2,…,tm∈G∖{1} and ϵ>0 there exists s1,…,sn∈G such that ‖1/ n∑ k=1 n λskt jsk −1‖<ϵ Clearly, this is an important characterization, because many previously study classes of c*-simple groups were always shown to satisfy at most minor variant of the latter property. In fact, it is nonetheless part of the original proof of powers that F2 is c*-simple. We prove in section 3 that the reduced crossed products over c*-simple groups satisfy a similar property. We also record that the above property is a group c*-algebra variant of the Dixmier property. A unital c*-algebra A is said to satisfy the Dixmier property if the closed convex hull of {uau¿|u∈U(A)} intersects the centre of A for all a∈A. Haagerup and Zsidó [30] proved that a unital, simple c*-algebraA always satisfies the Dixmier property, and that the intersection of the aforementioned closed convex hull and the centre always reduces to a point, if c*-algebra has a unique tracial state [30].
III. Recurrent subgroups. Independently, Kennedy [36] obtained an algebraic characterization of c*simplicity using the notion of recurrence for subgroups, hence a group-theoretical version of the topological dynamical notion of uniformly recurrent subgroup. A subgroup H of a group G is recurrent if there exists a final subset F⊆G∖{1} such that F∩gHg−1≠∅ for allg∈G. A discrete group is c* simple if and only if it has no amenable, recurrent subgroups. The proof of the implication of the infamous result (Theorem 1.7) requires generalization. We differ this until section 3 (Theorem 3.8). However, Theorem 1.7 partially settles the question of de la Harpe; whether there exist c*-simple groups without the unique trace property. Conversely, by composing the conditional expectation Cr ¿(G)→Cr ¿(R(G)) with trivial representation Cr ¿(R(G))→C (i.e., an existence result which follows from the amenability of R(G) [11, Theorem 2.6.8]), yields a state τ:Cr ¿(G)→C such that τ(λg)=1 for all g∈R(G). Since R(G) is normal, then for any two g,h∈G we have gh∈R(G) if and only if hg∈R(G), implying τ(λgλh)=τ(λhλg). Hence τ is a tracial state on Cr ¿(R(G)). The rest of this paper is organized as follows: In section 2, we give some preliminary results which we shall need later. In section 3, we proof our main results. Precisely, we proof Theorem 3.1, 3.2, 3.7, and 3.8. In section 4, we study stability properties that our results and many others in the literatures satisfy. OCLC -1091588944 X, i.e., φ|Z(A)=δx for some x∈X. Let g∈G∖{1}, then there exists f∈C(X) such that f(g−1x)≠f (x). This implies φ(λβ(g)(f (x))1H)=φ(λβ(g)f)=φ(gf λβ(g))=f (g−1x)φ(λβ(g)) Therefore,φ(λβ(g))=0. Let EA:A⋊α,r β G→A be the canonical conditional expectation, it follows thatφ=φ∘EA.

MAIN RESULTS
We now prove the following I. G II.
C(X)⋊rG , for some G-boundary X III.
C(X)⋊rG, for all G-boundary X IV.
The action of G on some X VI.
The action of G on ∂FG  10. The action of G on ∂FG is topologically free by Theorem 1.5, thus proving IV implies VI. Next, we need to prove I imply III. Let X be a G-boundary, using Lemma 2.6, we may assume that there is a G-equivariant unital c*-algebra inclusion C(X)⊆C(∂FG) . Letπ:C(∂FG)⋊rG→B be a unital *-homeomorphism. The action of G on ℬ may be defined by means of inner automorphisms Ad(π(λg)) of ℬ, so that π becomes Gequivariant. Using the inclusion C⊆C(X), we realizeCr ¿(G)as a unitalG-invariant c*subalgebra of C¿(X)⋊rG. If Cr ¿(G) is simple, then π|Cr ¿(G) is injective, so that canonical tracial state of τ:Cr ¿(G)→C⊆C(∂FG) extend to G-equivariant u.c.p map τ:B→C(∂FG) such that ~ τ∘π|Cr ¿(G)=τ by Lemma 2.7. Using Lemma 2.6 once again, we find that the map ~ τ∘π|C(X):C(X)→C(∂FG) is the inclusionmapC(X)→C(∂FG). Precisely, C(X)⊆mult(~ τ∘π). If E:C(X)⋊rG→C(X) is the canonical faithful conditional expectation, the ~ τ(π(f λg))=fτ(λg)=E(f λg) in C(∂FG) for all f∈C(X) and g∈G. Hence, ~ τ∘π=E, meaning that π is faithful and therefore injective. Henceforth, C(X)⋊rG is simple.Next, we need to prove II implies I. If C(X)⋊rG is simple for some Gboundary X, let I⊆Cr ¿(G) by a proper closed ideal. If φ:Cr ¿(G)→C is a state such that φ(I)={0} extend φ to a state on C(X)⋊rG and let (gi) be a net in G such that giμ→δx for some x∈X where μ=φ|C(X). By weak*-compactness we may assume that (φ∘Ad(λg)) converges to some state ψ on C(X)⋊rG, so that ψ|C(X)=δx and ψ|I=0. Thus, C(X)⊆mult(ψ).Furthermore, for any b∈I,f1f2∈C(∂FG) andf1f2∈G, It is not difficult to see that the ideal generated by I is proper. Therefore we have I={0} because C(∂FG)⋊rG was assumed to be simple.

Remark 3.2:
It follows from Theorem 3.1 that any c*-simple discrete group G has trivial amenable radical. Indeed, if the action of G on ∂FG is free, then R(G)=∩x∈∂FGGx={1} by Lemma 2.9. Since the result of Kalantar and Kennedy [90], other characterizations of c*-simplicity have been obtained (see, [4, 10, 29, 30, and 36] Assume that (A,G,α) is separable. If G is amenable, then every primitive ideal of A⋊rG is an induced primitive ideal. Moreover, if G acts freely on prim(A), then the induce process establishes a bijection between prim(A⋊rG) and the quasiorbits in prim(A). In particular, if G acts freely and every orbit is dense, then A⋊rG is simple. It is instructive to note that the twisted action and the equivalence of a group being c*-simple admits a free boundary action allows us to generalize many of these results. This we do in the following theorems. Proof: Let γ be a maximal G-invariant ideal in A. We claim that the ideal γ ´ ⋊α,r β G in A ´ ⋊α,r β G is maximal; assume thatJ is a proper ideal in A ´ ⋊α,r β G such that γ ´ ⋊α,r β G⊆J . Now, let (A⊗C(∂FG),G,ν,ι) of (A,G,α,β) be the natural extension.
Theorem 3.6 implies that K is proper, so the maximality of γ implies that γ=K∩A since K∩A is a Ginvariant. It follows that J⊆γ ´ ⋊α,r β G, and γ ´ ⋊α,r β G is maximal.We required an analysis to show that the ideal I∩A is maximal among proper Ginvariant ideals inA. Now let I be a maximal ideal inA ´ ⋊α,r β G. LetJ denote the ideals in ((A⊗C(∂FG)))⋊ν,r ι G generated byI. By Lemma 2.11J⊂JA ´ ⋊ν,r ι G whereJA=J∩(A⊗C(∂FG)). Hence by (2.3) I⊂J∩(A⋊α,r β G)⊂(JA ´ ⋊ν,r ι G)∩(A⋊α,r β G)=(J∩A) ´ ⋊α,r β G Since I is proper, Theorem 3.6 implies that J∩A is proper inA, whence maximality of I implies that I=(J∩A) ´ ⋊α,r β G. Now I∩A=J∩A follows from (2.2). It follows from our analysis I=(I∩A) ´ ⋊α,r β G(3.1) Now, let F be a proper G-invariant ideal in A such that I∩A⊂F. Then F ´ ⋊α,r β G is a proper ideal in A ´ ⋊α,r β G and I=(I∩A) ´ ⋊α,r β G⊂F ´ ⋊α,r β G.
Therefore the maximality of I implies that I=F ´ ⋊α,r β G. HenceI∩A=(F ´ ⋊α,r β G)∩A=F. Thus, I∩A is maximal. Finally, it now clear the correspondence is bijective follows from the identities (2.2) and (3.1) Corollary 3.3:Let (A,G,α,β) be a unital twisted C*-dynamical system where G is C*-simple. Then A⋊α,r β G is simple if and only if A is G-simple.

Corollary 3.4:
If G is C*-simple, then the reduced twisted group C*-algebra Cr ¿(G,β) is simple for every multiplier β:G×G→ℾ.
Corollary 3.5:Let (A,G,α,β) be as in Corollary 3.3. Let N be a normal subgroup of G. Write (α,β) for the restriction of (α,β) to N. If G/N is C*-simple, then A⋊α,r β G is simple whenever A⋊α,r β Nis simple.
Proof: A⋊α,r β G≌(A⋊α,r β N)⋊ν,r ι (G/N) follow from the existence of a twisted action (ι,ν) of G/NonA⋊α,r β N. The desire conclusion now follows from Corollary 3.3. It should be noted that the conclusion of Theorem 3.2 is not true if we allow the underlying c*-algebra to be nonunital. Thurs, c0(X) is always Gsimple, even though c0(X)⋊rG may contain many ideals. Furthermore, assume that G is a C*-simple group and A, a unital G-C*-algebra. If Z(I(A)) is G-simple, and A is prime, then the action of G on A has the intersection property OCLC -1091588944 C*-algebra by our analysis above. Thus, I ´ ⋊rG is a prime ideal of A⋊rG, therefore the map I↦I ´ ⋊rG is well defined, and it is injective since (I ´ ⋊rG)∩A=I for each Ginvariant ideal I⊆A.
Theorem 3.7:Let (A,G,α,β) be a unital twisted C*-dynamical system. Let X be a G -boundary and let (A⊗C(X),G,ν,ι) denote the associated natural extension. Let I be a proper ideal in A⋊α,r β G and let J denote the ideal in (A⊗C(X))⋊ν,r ι G generated by I. Then J is proper.

SOME EXAMPLES OF C*-DISCRETE GROUPS
We shall need the following Lemma.
Lemma 5.1: Let G be a Hausdorff topological group and let X be a minimal proximal compact G-space. If X has an isolated point, then X is a one point space. In all of the following examples, we have assume X to be boundaries that are not one-point spaces, such that X has no isolated points by Lemma 5.1. Precisely, finite subsets of X have empty interior.
Example I (Powers [42]):Non-abelian free groups of finite rank. For n≥2, the action of non-abelian free group Fn on its boundary ∂Fn of one-sided reduced infinite words is topological free. This implies that Fn is C*-simple. Indeed, if A is a free generating set for Fn, let Π=g1...gn∈Fn/{1} be a word in a reduced form, where g1...gn∈A∪A−1. We claim that XΠ is finite, so that it has empty interior. Taking the conjugation if necessary, assuming that g1gn≠1 since Π≠1, then gXΠ=XgΠg−1,g∈Fn so that XΠ has an empty interior if and only if XgΠg−1 has empty interior. If Πx=x for some x∈∂Fn, assume that the concatenation is reduced. Then the first n letter of x are the n letters of . Since g1gn≠1, Π2x is also reduced. Therefore, then next n letters of x are those of Π. On iterating this process, we see that x=ΠΠΠ . . . , if Πx is not reduced, then let 1≤k≤n be the largest such that the first k letters of x, are gn −1...gn−k+1 −1 . Since the first letter of x is gn −1 and the first letter of Πx is g1, assuming that k<n, we have k=n. Thus,