Deformations of nilpotent groups and homotopy symmetric $C^*$-algebras

Abstract

The homotopy symmetric $C^*$-algebras are those separable $C^*$-algebras for which one can unsuspend in E-theory. We find a new simple condition that characterizes homotopy symmetric nuclear $C^*$-algebras and use it to show that the property of being homotopy symmetric passes to nuclear $C^*$-subalgebras and it has a number of other significant permanence properties. As an application, we show that if $I(G)$ is the kernel of the trivial representation $\iota:C^*(G)\to \mathbb{C}$ for a countable discrete torsion free nilpotent group $G$, then $I(G)$ is homotopy symmetric and hence the Kasparov group $KK(I(G),B)$ can be realized as the homotopy classes of asymptotic morphisms $[[I(G),B \otimes \mathcal{K}]]$ for any separable $C^*$-algebra $B$.