ar X iv : m at h / 97 06 21 1 v 1 [ m at h . FA ] 6 J un 1 99 7 THE SIMILARITY DEGREE OF AN OPERATOR ALGEBRA

Let A be a unital operator algebra. Let us assume that every bounded unital homomorphism u: A → B(H) is similar to a contractive one. Let Sim(u) = inf{S S −1 } where the infimum runs over all invertible operators S: H → H such that the " conjugate " homomorphism a → S −1 u(a)S is contractive. Now for all c > 1, let Φ(c) = sup Sim(u) where the supremum runs over all unital homomorphism u: A → B(H) with u ≤ c. Then, there is α ≥ 0 such that for some constant K we have: (*) ∀c > 1 Φ(c) ≤ Kc α. Moreover, the smallest α for which this holds is an integer, denoted by d(A) (called the similarity degree of A) and (*) still holds for some K when α = d(A). Among the applications of these results, we give new characterizations of proper uniform algebras on one hand, and of nuclear C *-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.