Similarities for the maximal tensor product of certain C*-algebras

TL;DR

This paper investigates the Kadison's similarity property in the context of maximal tensor products of certain C*-algebras, establishing conditions under which the property is preserved and providing bounds on the similarity length.

Contribution

It proves that for unital C*-algebras satisfying Kadison's property, the maximal tensor product also satisfies the property with a bounded similarity length.

Findings

Kadison's similarity property is preserved under certain maximal tensor products.

The similarity length of the tensor product is bounded by the product of individual lengths and the tensor product's length.

Provides conditions ensuring the similarity property holds for tensor products of C*-algebras.

Abstract

We prove that if the unital $C^*$-algebras $\cl A$ and $\cl B$ satisfy Kadison's similarity property and the length $L=L\left(\cl A\tens\limits_{max}\cl B\right)$ of their maximal tensor product is finite, then $\cl A\tens\limits_{max}\cl \cl B$ satisfies Kadison's similarity property with similarity length $\ell\left(\cl A\tens\limits_{max}\cl B\right)\leq L \max\left\{\ell(\cl A),\,\ell(\cl B)\right\}.$