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

Abstract

We show that whenever m≥1 and M1,…,Mm are nonamenable factors in a large class of von Neumann algebras that we call C(AO) and which contains all free Araki–Woods factors, the tensor product factor M1⊗‾⋯⊗‾Mm retains the integer m and each factor Mi 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 M1,…,Mm 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 III1 factors in the class C(AO).