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Rigidity of free product von Neumann algebras. (English) Zbl 1379.46046
Summary: Let $$I$$ be any nonempty set and let $$(M_{i},\varphi_{i})_{i\in I}$$ be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class $${\mathcal{C}}_{\text{anti-free}}$$ of (possibly type $$\text{III}$$) von Neumann algebras including all nonprime factors, all nonfull factors and all factors possessing Cartan subalgebras. 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 $$\text{II}_{1}$$ factors and is new for free product type $$\text{III}$$ factors. It moreover provides new rigidity phenomena for type $$\text{III}$$ factors.

##### MSC:
 46L10 General theory of von Neumann algebras 46L09 Free products of $$C^*$$-algebras 46L36 Classification of factors
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