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arXiv · 1507.02157

Rigidity of free product von Neumann algebras

Cyril Houdayer·Yoshimichi Ueda
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Abstract

Let $I$ be any nonempty set and $(M_i, \varphi_i)_{i \in I}$ any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class $\mathcal C_{\rm anti-free}$ 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 $(M, \varphi) = \ast_{i \in I} (M_i, \varphi_i)$, 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.

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Cyril Houdayer, Yoshimichi Ueda. 2015-07-08. Rigidity of free product von Neumann algebras. https://doi.org/10.1112/s0010437x16007673

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