Rigidity of free product von Neumann algebras

Abstract

Let $I$ be any nonempty set and $(M_i,{\phi }_i{)}_{i\in I}$ any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class ${\mathcal{C}}_{\mathrm{anti}-\mathrm{free}}$ of (possibly type III) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing a Cartan subalgebra. For the free product $(M,\phi )={\ast }_{i\in I}(M_i,{\phi }_i)$, we show that the free product von Neumann algebra $M$ retains the cardinality $|I|$ and each nonamenable factor $M_i$ up to stably inner conjugacy, after permutation of the indices. Our main theorem unifies all previous Kurosh-type rigidity results for free product type II${}_1$ factors and is new for free product type III factors. It moreover provides new rigidity phenomena for type III factors.

30 pages

References in corpus (3)

• Unique prime factorization and bicentralizer problem for a class of type III factors
• Asymptotic structure of free product von Neumann algebras
• Asymptotic structure of free Araki-Woods factors

Cited by in corpus (8)

• Unique prime factorization and bicentralizer problem for a class of type III factors
• W*-rigidity paradigms for embeddings of II${}_1$ factors
• Fullness and Connes' $\mathbf{\tau }$ invariant of type III tensor product factors
• Solidity of type III Bernoulli crossed products
• Structure of extensions of free Araki-Woods factors
• Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras
• Boundary and rigidity of nonsingular Bernoulli actions
• Product rigidity in von Neumann and C${}^{\ast }$-algebras via s-malleable deformations