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 $M_1\overline{\otimes }\cdots \overline{\otimes }{M}_m$ 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 [OP03, Is14] 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 $\mathrm{II}{\mathrm{I}}_1$ factors in the class ${\mathcal{C}}_{(\text{AO})}$.

41 pages. v3: final version

References in corpus (3)

• Asymptotic structure of free Araki-Woods factors
• Rigidity of free product von Neumann algebras
• Gamma stability in free product von Neumann algebras

Cited by in corpus (25)

• Asymptotic structure of free product von Neumann algebras
• Amenable absorption in amalgamated free product von Neumann algebras
• Rigidity of free product von Neumann algebras
• Strong solidity of free Araki-Woods factors
• Classification of a family of non almost periodic free Araki-Woods factors
• Structure of bicentralizer algebras and inclusions of type III factors
• Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups
• On fundamental groups of tensor product $\mathrm{I}{\mathrm{I}}_1$ factors
• Fullness and Connes' $\mathbf{\tau }$ invariant of type III tensor product factors
• Solidity of type III Bernoulli crossed products
• Connes' bicentralizer problem for q-deformed Araki-Woods algebras
• Structure of extensions of free Araki-Woods factors
• Prime II${}_1$ factors arising from irreducible lattices in products of rank one simple Lie groups
• Rigidity for von Neumann algebras given by locally compact groups and their crossed products
• Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras
• Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors
• Boundary and rigidity of nonsingular Bernoulli actions
• Unique prime factorization for infinite tensor product factors
• Crossed-products by locally compact groups: Intermediate subfactors
• Classification of Tensor Decompositions of II${}_1$ Factors Associated With Poly-Hyperbolic Groups
• On the relative bicentralizer flows and the relative flow of weights of inclusions of factors of type III${}_1$
• A free product pair rigidity result in von Neumann algebras
• Ozawa's class $\mathcal{S}$ for locally compact groups and unique prime factorization
• Classification of tensor decompositions for II${}_1$ factors
• Classification results for nonsingular Bernoulli crossed products