Discrete groups and simple C*-algebras

Abstract

Let G denote a discrete group and let us say that G is C*-simple if the reduced group C*-algebra associated with G is simple. We notice immediately that there is no interest in considering here the full group C*-algebra associated with G, because it is simple if and only if C is trivial. Since Powers in 1975 [26] proved that all nonabelian free groups are C*-simple, the class of C*-simp1e groups has been considerably enlarged (see [1, 2, 6, 7, 12, 13, 14, 16, 24] as a sample!), and two important subclasses are so-called weak Powers groups ([6, 13]; see Section 4 for definition and examples) and the groups of Akemann-Lee type [1, 2], which are groups possessing a normal non-abelian free subgroup with trivial centralizer.

Approximation Properties for Crossed Products by Actions and Coactions

“…There is a naturally induced action α of G on C * r (H), and the crossed product C * r (H) × α,r G is isomorphic to the reduced C * -algebra C * r (H × ρ G) of the semi-direct product H × ρ G, [2]. It was established in [9] that, for the natural action of SL(2, Z) on Z 2 , Λ(C * r (Z 2 × ρ SL(2, Z))) = ∞, while Λ(C * r (Z 2 )) and Λ(C * r (SL(2, Z)) are both finite.…”

On twisted group -algebras associated with FC-hypercentral groups and other related groups

“…J. Packer [22] has shown that all countable nilpotent groups belong to K am (see also [26] for the case of abelian groups). Large families of nonamenable groups in K are described in [1,2,4]. We mention explicitly the class P introduced in [4], as we will refer to it later: it consists of all PH groups [24] and of all groups with property (P com ) [5].…”

A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups

“…When the cocycle is trivial (σ(s, t) = 1), this is just the usual reduced C * -crossed product. If we replace A by a von Neumann algebra M, then the w * -closure of the algebra generated by these operators is the twisted von Neumann algebra, which we denote by M ⋊ σ α,vn G to avoid confusion with the full twisted crossed product [28], usually written as A ⋊ σ α G. As was mentioned in the introduction, twisted crossed products arise when expressing A ⋊ α,r G as (A ⋊ α,r N) ⋊ σ β,r G/N, where N is a normal subgroup of G. We digress briefly to show how this is achieved (see [1]). Specify a cross section φ : G/N → G, and define β s to be Ad φ(s) for s ∈ G/N, noting that β s maps A ⋊ α,r N to itself since N is a normal subgroup.…”

“…When N is a normal subgroup of G, it is desirable to decompose A ⋊ α,r G as an iterated crossed product by an action of N and then by an action of the quotient group G/N on A ⋊ α,r N. However, this is only possible in general when we expand to the larger class of reduced twisted crossed products with the extra ingredient of a cocycle, a map σ of G × G into the unitary group U(A) of A, (see [1,Theorem 2.1]). We show that our results on reduced crossed products extend to include the twisted case.…”