Gamma stability in free product von Neumann algebras

Abstract

Let $(M,\phi )=(M_1,{\phi }_1)\ast (M_2,{\phi }_2)$ be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer $M_1^{{\phi }_1}$ is diffuse. We first show that any intermediate subalgebra $M_1\subset Q\subset M$ which has nontrivial central sequences in $M$ is necessarily equal to $M_1$. Then we obtain a general structural result for all the intermediate subalgebras $M_1\subset Q\subset M$ with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III${}_1$ factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras.

arXiv comment18 pages

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Cited by in corpus (7)

• 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
• Structure of modular invariant subalgebras in free Araki-Woods factors
• Asymptotic freeness in tracial ultraproducts
• Structure of extensions of free Araki-Woods factors
• Consequences of the random matrix solution to the Peterson-Thom conjecture