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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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