Simultaneous similarity, bounded generation and amenability

Abstract

We prove that a discrete group $G$ is amenable if and only if it is strongly unitarizable in the following sense: every unitarizable representation $\pi$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $\pi$. Analogously, a $C^*$-algebra $A$ is nuclear if and only if any bounded homomorphism $u: A\to B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $\xi$ in the von Neumann algebra generated by the range of $u$ such that $a\to \xi u(a) \xi^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations. We apply this to answer several questions left open in our previous work concerning the length $L(A\otimes_{\max} B)$ of the maximal tensor product $A\otimes_{\max} B$ of two unital $C^*$-algebras, when we consider its generation by the subalgebras $A\otimes 1$ and $1\otimes B$. We show that if $L(A\otimes_{\max} B)<\infty$ either for $B=B(\ell_2)$ or when $B$ is the $C^*$-algebra (either full or reduced) of a non-Abelian free group, then $A$ must be nuclear. We also show that $L(A\otimes_{\max} B)\le d$ if and only if the canonical quotient map from the unital free product $A\,{\ast}\, B$ onto $A\otimes_{\max} B$ remains a complete quotient map when restricted to the closed span of the words of length at most $d$.