# A Similarity Degree Characterization of Nuclear $C_{∗}$-algebras

### Gilles Pisier

Texas A&M University, College Station, USA

## 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.

## Cite this article

Gilles Pisier, A Similarity Degree Characterization of Nuclear $C_{∗}$-algebras. Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, pp. 691–704

DOI 10.2977/PRIMS/1166642155