The similarity degree of an operator algebra

Abstract

Let $A$ be a unital operator algebra. Let us assume that every {\it bounded\/} unital homomorphism $u:A\to B(H)$ is similar to a {\it contractive\/} one. Let $\text{Sim}(u)=\inf \{\Vert S\Vert \Vert S^{-1}\Vert \}$ where the infimum runs over all invertible operators $S:H\to H$ such that the ``conjugate'' homomorphism $a\to S^{-1}u(a)S$ is contractive. Now for all $c>1$, let $\Phi (c)=\sup \text{Sim}(u)$ where the supremum runs over all unital homomorphism $u:A\to B(H)$ with $\Vert u\Vert \le c$. Then, there is $\alpha \ge 0$ such that for some constant $K$ we have: $$Φ(c) \le Kc^α.\leqno (*)\qquad \forall c>1$$ Moreover, the smallest $\alpha$ for which this holds is an integer, denoted by $d(A)$ (called the similarity degree of $A$) and $(\ast )$ still holds for some $K$ when $\alpha =d(A)$. Among the applications of these results, we give new characterizations of proper uniform algebras on one hand, and of nuclear $C^{\ast }$-algebras on the other. Moreover, we obtain a characterization of amenable groups which answers (at least partially) a question on group representations going back to a 1950 paper of Dixmier.

Cited by in corpus (1)

• A non-commutative Amir-Cambern theorem for von Neumann algebras and nuclear $C^{\ast }$-algebras