A similarity degree characterization of nuclear $C^*$-algebras

Abstract

We show that a $C^*$-algebra $A$ is nuclear iff there is a constant $K$ and $α<3$ such that, for any bounded homomorphism $u\colon A \to B(H)$, there is an isomorphism $ξ\colon H\to H$ satisfying $\|ξ^{-1}\|\|ξ\| \le K\|u\|^α$ and such that $ ξ^{-1} u(.) ξ$ is a $*$-homomorphism. In other words, an infinite dimensional $A$ is nuclear iff its length (in ths sense of our previous work on the Kadison similarity problem) is equal to 2.