Deformations of wreath products

Abstract

Connectivity is a homotopy invariant property of a separable C * -algebra A which has three important consequences: absence of nontrivial projections, quasidiagonality and realization of the Kasparov group KK(A, B) as homotopy classes of asymptotic morphisms from A to B ⊗ K if A is nuclear. Here we give a new characterization of connectivity for separable exact C*-algebras and use this characterization to show that the class of discrete countable amenable groups whose augmentation ideals are connective is closed under generalized wreath products. In a related circle of ideas, we give a result on quasidiagonality of reduced crossed-product C*-algebras associated to noncommutative Bernoulli actions.

Examples of non connective C *-algebras

“…The group G is called diffuse if every non-empty finite subset A of G has an element a ∈ A such that for any g ∈ G, either ga or g −1 a is not in A, see [6] and [7]. More examples of nonabelian connective groups were exhibited in [4], [5] and [6]. Dadarlat's group G 2 is a 3-dimensional, torsion-free crystallographic group, where a crystallographic group Γ, of dimension n is a cocompact and discrete subgroup of the isometry group E(n) = O(n) R n of the Euclidean space R n .…”

Obstructions to matricial stability of discrete groups and almost flat K-theory

“…(ii) If H is quasidiagonal and G is amenable, then the wreath product H ≀ G is quasidiagonal. Indeed, by [31,Thm.4.2], if D is a unital separable quasidiagonal C * -algebra and G is a countable discrete amenable group, then the crossed product ( G D) ⋊ G is quasidiagonal (here we work with minimal tensor products and G acts via noncommutative Bernoulli shifts). Thus, if there is an embedding ω :…”

Connective Bieberbach groups

“…If G is connective and amenable, then K 0 (C * (G)) = Z[ι] ⊕ [[I(G), K]]. This implies that the nontrivial part of the K-homology of C * (G) can be realized as homotopy classes of asymptotic representations {π t : G → U (∞)} t∈ [1,∞) with t → π t (g) continuous, g ∈ G, and lim t→∞ π t (g 1 g 2 ) − π t (g 1 )π t (g 2 ) = 0, g 1 , g 2 ∈ G. Large classes of amenable connective groups were exhibited in [5], [6], [7].…”

“…is connective since by[7, Cor. 3.3] direct products of discrete amenable connective groups are connective.…”