Simultaneous similarity, bounded generation and amenability

摘要

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 -> 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 it such that a -> 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 circle times(max) B) of the maximal tensor product A circle times(max) B of two unital C*-algebras, when we consider its generation by the subalgebras A circle times 1 and 1 circle times B. We show that if L(A circle times(max) B) < infinity either for B = B(l(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 circle times(max) B) <= d if and only if the canonical quotient map from the unital free product A * B onto A circle times(max) B remains a complete quotient map when restricted to the closed span of the words of length at most d.