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

Let $Γ$ be a discrete group. We show that if $Γ$ is nonamenable, then the algebraic tensor products $C^*_r(Γ)\otimes C^*_r(Γ)$ and $C^*(Γ)\otimes C^*_r(Γ)$ do not admit unique $C^*$-norms. Moreover, when $Γ_1$ and $Γ_2$ are discrete groups containing copies of noncommutative free groups, then $C^*_r(Γ_1)\otimes C^*_r(Γ_2)$ and $C^*(Γ_1)\otimes C_r^*(Γ_2)$ 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.

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