A Similarity Degree Characterization of Nuclear $C^{\ast }$-algebras

Abstract

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