A similarity degree characterization of nuclear $C^{\ast }$-algebras

Abstract

We show that a $C^{\ast }$-algebra $A$ is nuclear iff there is a constant $K$ and $\alpha <3$ 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 ths sense of our previous work on the Kadison similarity problem) is equal to 2.

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