A Similarity Degree Characterization of Nuclear <em>C</em>^∗-algebras

Abstract

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‖ ≤ K ‖u‖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.