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 decompositions of ${\mathrm{II}}_1$ factors of the form $M\bar{\otimes }N$, where $M$ is a non-McDuff factor.

8 pages, to appear in Publications of RIMS, v3: improved exposition, Corollary 3.5 and comments on Corollary 3.4 and Corollary 3.5 added, v2: improved exposition, e-mail address changed on page 8