# A Similarity Degree Characterization of Nuclear <em>C</em><sup>∗</sup>-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 <em>C</em><sup>∗</sup>-algebras. Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, pp. 691–704

DOI 10.2977/PRIMS/1166642155