Simultaneous similarity, bounded generation and amenability

Details Simultaneous similarity, bounded generation and amenability Simultaneous similarity, bounded generation and amenability

2005-08-12

arxiv.org/abs/math/0508223v2

Mathematics

Operator Algebras

There are several improvements both to the exposition and to some results. The main refinement is that if the length of the mi

Scientific paper

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 $\le d$.

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