Simultaneous Similarity, Bounded Generation and Amenability

We prove that a discrete group G is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation π on G can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of π. Analogously a C *-algebra A is nuclear iff any bounded homomorphism u : A → B(H) is strongly similar to a *-homomorphism in the sense that there is an invertible operator ξ in the von Neumann algebra generated by the range of u such that a → ξu(a)ξ −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, 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, B) < ∞ 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, B) ≤ d iff 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 ≤ d.