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

Abstract

Let $\Gamma$ be a discrete group. We show that if $\Gamma$ is nonamenable, then the algebraic tensor products $C_r^{\ast }(\Gamma )\otimes C_r^{\ast }(\Gamma )$ and $C^{\ast }(\Gamma )\otimes C_r^{\ast }(\Gamma )$ do not admit unique $C^{\ast }$-norms. Moreover, when ${\Gamma }_1$ and ${\Gamma }_2$ are discrete groups containing copies of noncommutative free groups, then $C_r^{\ast }({\Gamma }_1)\otimes C_r^{\ast }({\Gamma }_2)$ and $C^{\ast }({\Gamma }_1)\otimes C_r^{\ast }({\Gamma }_2)$ admit $2^{{\aleph }_0}$ $C^{\ast }$-norms. Analogues of these results continue to hold when these familiar group $C^{\ast }$-algebras are replaced by appropriate intermediate group $C^{\ast }$-algebras.

6 pages, v2 contains minor changes and an extra remark. To appear in Bulletin of the London Mathematical Society

References in corpus (1)

• Exotic group C*-algebras

Cited by in corpus (1)

• $L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups