C*-norms for tensor products of discrete group C*-algebras

TL;DR

This paper investigates the uniqueness of C*-norms on tensor products of group C*-algebras, showing non-uniqueness for nonamenable groups and establishing the existence of many norms for groups containing free subgroups.

Contribution

It demonstrates non-uniqueness of C*-norms for tensor products involving nonamenable groups and shows the abundance of norms when groups contain free subgroups.

Findings

Nonamenable groups lead to non-unique C*-norms on tensor products.

Groups with free subgroups admit continuum many C*-norms.

Results extend to intermediate group C*-algebras.

Abstract

Let $\Gamma$ be a discrete group. We show that if $\Gamma$ is nonamenable, then the algebraic tensor products $C^*_r(\Gamma)\otimes C^*_r(\Gamma)$ and $C^*(\Gamma)\otimes C^*_r(\Gamma)$ do not admit unique $C^*$-norms. Moreover, when $\Gamma_1$ and $\Gamma_2$ are discrete groups containing copies of noncommutative free groups, then $C^*_r(\Gamma_1)\otimes C^*_r(\Gamma_2)$ and $C^*(\Gamma_1)\otimes C_r^*(\Gamma_2)$ admit $2^{\aleph_0}$ $C^*$-norms. Analogues of these results continue to hold when these familiar group $C^*$-algebras are replaced by appropriate intermediate group $C^*$-algebras.