A Similarity Degree Characterization of Nuclear $C^∗$-algebras

We show that a $C^*$-algebra $A$ is nuclear iff there is a constant $K$ and $\alpha<3$ such that, for any bounded homomorphism $u\colon A \to B(H)$, there is an isomorphism $\xi\colon H\to H$ satisfying $\|\xi^{-1}\|\|\xi\| \le K\|u\|^\alpha$ and such that $ \xi^{-1} u(.) \xi$ 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.