Simultaneous similarity, bounded generation and amenability
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We prove that a discrete group G is amenable if and only if it is strongly unitarizable in the following sense: every unitarizable representation on G can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of . Analogously, a C*-algebra A is nuclear if and only if any bounded homomorphism u : A B(H) is strongly similar to a *-homomorphism in the sense that there is an invertible operator in the von Neumann algebra generated by the range of u such that a u(a)-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 max B) of the maximal tensor product A max B of two unital C*-algebras, when we consider its generation by the subalgebras A 1 and 1 B. We show that if L(A max B) < either for B = B(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 max B) d if and only if the canonical quotient map from the unital free product A * B onto A max B remains a complete quotient map when restricted to the closed span of the words of length at most d.