Simultaneous similarity, bounded generation and amenability

We prove that a discrete group $G$ is amenable iff 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 $C^{\ast }$-algebra $A$ is nuclear iff any bounded homomorphism $u:A\to B(H)$ is strongly similar to a $\ast$-homomorphism in the sense that there is an invertible operator $\xi$ in the von Neumann algebra generated by the range of $u$ such that $a\to \xi u(a) \xi^{-1}$isa$*$-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 $\mathrm{\ell }(A,B)$ of the maximal tensor product $A{\otimes }_{\max }B$ of two unital $C^{\ast }$-algebras, when we consider its generation by the subalgebras $A\otimes 1$ and $1\otimes B$. We show that if $\mathrm{\ell }(A,B)<\mathrm{\infty }$ either for $B=B({\mathrm{\ell }}_2)$ or when $B$ is the $C^{\ast }$-algebra (either full or reduced) of a non Abelian free group, then $A$ must be nuclear. We also show that $\mathrm{\ell }(A,B)\le d$ iff 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 $\le d$.