Simultaneous similarity, bounded generation and amenability

الأصل · EN

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⊗ 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(ℓ₂) 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⊗ B remains a complete quotient map when restricted to the closed span of the words of length ≤ d.