Similarities for the maximal tensor product of certain 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 $\ell\left(\cl A\tens\limits_{max}\cl B\right)\leq L \max\left\{\ell(\cl A),\,\ell(\cl B)\right\}.$