Asymptotic structure of free product von Neumann algebras

Abstract

Let $(M,\phi )=(M_1,{\phi }_1)\ast (M_2,{\phi }_2)$ be the free product of any $\sigma$-finite von Neumann algebras endowed with any faithful normal states. We show that whenever $Q\subset M$ is a von Neumann subalgebra with separable predual such that both $Q$ and $Q\cap M_1$ are the ranges of faithful normal conditional expectations and such that both the intersection $Q\cap M_1$ and the central sequence algebra $Q^{\prime }\cap M^{\omega }$ are diffuse (e.g. $Q$ is amenable), then $Q$ must sit inside $M_1$. This result generalizes the previous results of the first named author in [Ho14] and moreover completely settles the questions of maximal amenability and maximal property Gamma of the inclusion $M_1\subset M$ in arbitrary free product von Neumann algebras.

26 pages. v3: final version

References in corpus (4)

• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• Unique prime factorization and bicentralizer problem for a class of type III factors
• Asymptotic structure of free Araki-Woods factors
• Gamma stability in free product von Neumann algebras

Cited by in corpus (14)

• Amenable absorption in amalgamated free product von Neumann algebras
• Strong solidity of free Araki-Woods factors
• Rigidity of free product von Neumann algebras
• Structure of bicentralizer algebras and inclusions of type III factors
• Structure of modular invariant subalgebras in free Araki-Woods factors
• Solidity of type III Bernoulli crossed products
• Structure of extensions of free Araki-Woods factors
• Asymptotic freeness in tracial ultraproducts
• Connes' bicentralizer problem for q-deformed Araki-Woods algebras
• 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
• Unique prime factorization for infinite tensor product factors
• A free product pair rigidity result in von Neumann algebras
• Maximal amenable MASAs of the free group factor of two generators arising from the free products of hyperfinite factors