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Tohoku Mathematical Journal
2007
March
SECOND SERIES VOL. 59, NO. 1
Tohoku Math. J.
59 (2007), 7999

Title
SIMULTANEOUS SIMILARITY, BOUNDED GENERATION AND AMENABILITY
Author
Gilles Pisier
(Received August 26, 2005, revised June 19, 2006) 
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 nonAbelian 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 $d$.
2000 Mathematics Subject Classification.
Primary 46L06; Secondary 46L07.


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