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

Abstract

We show that the twisted group C * -algebra associated with a discrete FChypercentral group is simple (resp. has a unique tracial state) if and only if Kleppner's condition is satisfied. This generalizes a result of J. Packer for countable nilpotent groups. We also consider a larger class of groups, for which we can show that the corresponding reduced twisted group C * -algebras have a unique tracial state if and only if Kleppner's condition holds.MSC 2010: Primary: 46L55. Secondary: 22D25, 20C25, 46L05.

On reduced twisted group C*-algebras that are simple and/or have a unique trace

“…Since the class of C * -simple groups is (strictly) contained in the class of groups with the unique trace property [15,13], we get that every C * -simple group belongs to K, while every group with the unique trace property belongs to K U T . Combining results from [7] and [17], we show in the present paper that a group G belongs to K U T whenever the FC-hypercenter of G coincides with its amenable radical (cf. Theorem 3.7).…”

“…Instead of using F C(G) as the "base case" in the inductive proof of this theorem, we replace it by the (larger) normal subgroup N . Then the same proof as in [7] will work, provided that G/N is FC-hypercentral and N (and thus G) is amenable. We leave the details to the reader.…”

“…A useful consequence of a result due to Murphy [40] is the following theorem (cf. Corollaries 2.3 and 2.4 in [7]): Theorem 2.1. Assume that G is amenable, or that G is exact and A = C * r (G, σ) has stable rank one (i.e., the invertible elements of A are dense in A).…”

“…U T (G) = {σ ∈ Z 2 (G, T) | (G, σ) has the unique trace property}, K(G) = {σ ∈ Z 2 (G, T) | σ satisfies Kleppner's condition}, we always have C * S(G) ⊂ K(G) and U T (G) ⊂ K(G). Following [7], we will let K C * S (resp. K U T ) denote the class of groups G satisfying C * S(G) = K(G) (resp.…”

Fourier theory andC∗-algebras

“…One may of course also study simplicity and uniqueness of the trace for twisted reduced group C * -algebras, and quite a lot is known. We refer to [84,85] for an overview on this theme, as well as some new results.…”