On Central Sequence Algebras of Tensor Product von Neumann Algebras

<jats:p> We show that when <jats:inline-formula> <jats:tex-math>M</jats:tex-math> </jats:inline-formula> , <jats:inline-formula> <jats:tex-math>N_{1}</jats:tex-math> </jats:inline-formula> , <jats:inline-formula> <jats:tex-math>N_{2}</jats:tex-math> </jats:inline-formula> are tracial von Neumann algebras with <jats:inline-formula> <jats:tex-math>M'\cap M^{\omega}</jats:tex-math> </jats:inline-formula> abelian, <jats:inline-formula> <jats:tex-math>M'\cap(M\bar \otimes N_{1})^{\omega}</jats:tex-math> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math>M'\cap(M\bar \otimes N_{2})^{\omega}</jats:tex-math> </jats:inline-formula> commute in <jats:inline-formula> <jats:tex-math>(M\bar \otimes N_{1}\bar \otimes N_{2})^{\omega}</jats:tex-math> </jats:inline-formula> . As a consequence, we obtain information on McDuff decomposition of II <jats:inline-formula> <jats:tex-math>_{1}</jats:tex-math> </jats:inline-formula> factors of the form <jats:inline-formula> <jats:tex-math>M\bar \otimes N</jats:tex-math> </jats:inline-formula> , where <jats:inline-formula> <jats:tex-math>M</jats:tex-math> </jats:inline-formula> is a non-McDuff factor. </jats:p>