On central sequence algebras of tensor product von Neumann algebras

We show that when $M,N_{1},N_{2}$ are tracial von Neumann algebras with $M'\cap M^{\omega}$ abelian, $M'\cap(M\bar{\otimes}N_{1})^{\omega}$ and $M'\cap(M\bar{\otimes}N_{2})^{\omega}$ commute in $(M\bar{\otimes}N_{1}\bar{\otimes}N_{2})^{\omega}$. As a consequence, we obtain information on McDuff decompositions of $\rm{II}_{1}$ factors of the form $M\bar{\otimes}N$, where $M$ is a non-McDuff factor.