Unique prime factorization and bicentralizer problem for a class of type III factors

Abstract

We show that whenever $m\ge 1$ and $M_1,\dots ,M_m$ are nonamenable factors in a large class of von Neumann algebras that we call ${\mathcal{C}}_{(\text{AO})}$ and which contains all free Araki–Woods factors, the tensor product factor $M_1\bar{\otimes }\cdots \bar{\otimes }{M}_m$ retains the integer m and each factor $M_i$ up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [33], [25] and moreover provides new UPF results in the case when $M_1,\dots ,M_m$ are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type $\mathrm{I}\mathrm{I}{\mathrm{I}}_{\mathrm{1}}$ factors in the class ${\mathcal{C}}_{(\text{AO})}$.