# Simultaneous similarity, bounded generation and amenability

Research paper by **Gilles Pisier**

Indexed on: **16 Sep '05**Published on: **16 Sep '05**Published in: **Mathematics - Operator Algebras**

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#### Abstract

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 $C^*$-algebra $A$ is nuclear iff
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 $\ell(A,B)$ 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
$\ell(A,B)<\infty$ 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 $\ell(A,B)\le d$ 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 $\le d$.