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

TL;DR

This paper characterizes homotopy symmetric nuclear $C^*$-algebras with a new simple condition, demonstrating their permanence properties and applying this to show certain ideals of nilpotent group $C^*$-algebras are homotopy symmetric.

Contribution

It introduces a new simple criterion for homotopy symmetry in nuclear $C^*$-algebras and explores its implications and permanence properties.

Findings

Homotopy symmetry passes to nuclear subalgebras.

The property is stable under certain algebraic operations.

Ideals associated with nilpotent groups are homotopy symmetric.

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 $B$.