A Similarity Degree Characterization of Nuclear C * -algebras

We show that a C *-algebra A is nuclear iff there is a number α < 3 and a constant K such that, for any bounded homomorphism u : A → B(H), there is an isomorphism ξ : H → H satisfying ξ −1 ξ ≤ Ku α and such that ξ −1 u(.)ξ is a *-homomorphism. In other words, an infinite dimensional A is nuclear iff its length (in the sense of our previous work on the Kadison similarity problem) is equal to 2. Let A, B be C *-algebras. Let α be a C *-norm on their algebraic tensor product, denoted by A ⊗ B; as usual, A ⊗ α B then denotes the C *-algebra obtained by completing A ⊗ B with respect to α. By classical results (see [21]) the set of C *-norms admits a minimal and a maximal element denoted respectively by · min and · max. Then A is called nuclear if for any B we have A ⊗ min B = A ⊗ max B, or equivalently x min = x max for any x in A ⊗ B. We refer the reader to [21, 12] for more information on nuclear C *-algebras. We note in particular that by results due to Connes and Haagerup ([6, 7]), a C *-algebra is nuclear iff it is amenable as a Banach algebra (in B.E. Johnson's sense). The main result of this note is as follows. Theorem 1. The following properties of C *-algebra A are equivalent: (i) A is nuclear. (ii) There are α < 3 and a constant K such that any bounded homomorphism u : A → B(H) satisfies u cb ≤ Ku α. (iii) Same as (ii) with K = 1 and α = 2. The implication (i) ⇒ (iii) is well known (see [1, 3]). In the terminology of [15], the similarity degree d(A) is the smallest α for which the property considered in (ii) above is satisfied. It is proved in [15] that d(A) is always an integer identified as the smallest length of a specific kind of factorization for matrices with entries in A. With this terminology, the preceding theorem means that A is nuclear iff d(A) ≤ 2. In the infinite dimensional case, d(A) > 1 hence A is nuclear iff d(A) = 2.