A criterion for splitting $C^\ast $-algebras in tensor products

Abstract

The goal of the paper is to prove the following theorem: if $A$,$D$ are unital $C^{*}$-algebras, $A$ simple and nuclear, then any $C^{*}$-subalgebra of the $C^{*}$-tensor product of $A$ and $D$, which contains the tensor product of $A$ with the scalar multiples of the unit of $D$, splits in the $C^{*}$-tensor product of $A$ with some $C^{*}$-subalgebra of $D$.