Connective C⁎-algebras

Abstract

Connectivity is a homotopy invariant property of separable C * -algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C * -algebras using asymptotic morphisms. The purpose of this paper is to further explore the class of connective C *algebras. We give new characterizations of connectivity for exact and for nuclear separable C * -algebras and show that an extension of connective separable nuclear C * -algebras is connective. We establish connectivity or lack of connectivity for C * -algebras associated to certain classes of groups: virtually abelian groups, linear connected nilpotent Lie groups and linear connected semisimple Lie groups.

Connective Bieberbach groups

“…Proof of Theorem 1.2 (i) ⇒ (ii) Suppose that G is a connective Bieberbach group. Then all its subgroups are connective, since connectivity passes to subgroups [5]. It follows then by Theorem 1.1 and Proposition 2.1 that every nontrivial subgroup of G has a nontrivial center.…”

“…Connectivity [5] is a homotopy invariant property of a separable C * -algebra A that has three interesting consequences: absence of nonzero projections, quasidiagonality, and realization of the Kasparov groups as homotopy classes of asymptotic morphisms from A to B ⊗ K without suspensions, that is KK(A, B) ∼ = [[A, B ⊗ K]], if A is nuclear. A countable discrete group G is called connective if the kernel I(G) of the trivial representation ι : C * (G) → C is a connective C * -algebra.…”

“…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].…”

“…Connectivity of a separable nuclear C * -algebra A can be characterized solely in terms of its primitive spectrum Prim(A). It was shown in [5] that the existence of a nonempty compact open subset of Prim(A) is an obstruction to connectivity. By a remarkable result of J. Gabe [10] this is the only obstruction.…”

“…The orbit space R k /G of a Bieberbach group is a k-dimensional closed flat Riemannian manifold M with holonomy group isomorphic to D. Which Bieberbach groups are connective? It was shown in [5] that while any Bieberbach group with cyclic holonomy is connective [5,Thm. 3.8], the celebrated Hantzsche-Wendt group [11] (denoted here by Γ) is not connective [5,Cor.…”