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

We prove that if the unital $C^{\ast }$-algebras $\text{cl}A$ and $\text{cl}B$ 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\}.