Property $T$ of reduced $C^*$-crossed products by discrete groups

TL;DR

This paper extends the understanding of property $T$ in reduced $C^*$-crossed products, showing that such property implies the underlying group is finite and the algebra is finite dimensional, with applications to uniform Roe algebras.

Contribution

It generalizes previous results by Kamalov, establishing new conditions under which property $T$ implies finiteness of the group and algebra.

Findings

If $A\rtimes_{\alpha,r} G$ has property $T$, then $G$ is finite and $A$ is finite dimensional.

An infinite discrete group $H$ is non-amenable iff $C^*_u(H)$ has property $T$.

The result links property $T$ of crossed products to group finiteness and non-amenability.

Abstract

We generalize the main result of Kamalov and show that if $G$ is an amenable discrete group with an action $\alpha$ on a finite nuclear unital $C^*$-algebra $A$ such that the reduced crossed product $A\rtimes_{\alpha,r} G$ has property $T$, then $G$ is finite and $A$ is finite dimensional. As an application, an infinite discrete group $H$ is non-amenable if and only if the uniform Roe algebra $C^*_u(H)$ has property $T$.