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 $M^{\prime }\cap M^{\omega }$ abelian, $M^{\prime }\cap (M\bar{\otimes }{N}_1{)}^{\omega }$ and $M^{\prime }\cap (M\bar{\otimes }{N}_2{)}^{\omega }$ 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.