On Central Sequence Algebras of Tensor Product von Neumann Algebras

Abstract

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 decomposition of II$\_{1}$ factors of the form $M\bar \otimes N$, where $M$ is a non-McDuff factor.