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Bass-Serre rigidity results in von Neumann algebras. (English) Zbl 1201.46057
Summary: We obtain new Bass-Serre-type rigidity results for II\(_1\) equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any nonamenable factor arising as an amalgamated free product of von Neumann algebras \(\mathcal{M}_1 \ast_B \mathcal{M}_2\) over an abelian von Neumann algebra \(B\) is prime, that is, cannot be written as a tensor product of diffuse factors. This gives, both in the type II\(_1\) and in the type III cases, new examples of prime factors.

MSC:
46L36 Classification of factors
46L10 General theory of von Neumann algebras
46L54 Free probability and free operator algebras
46L55 Noncommutative dynamical systems
46L09 Free products of \(C^*\)-algebras
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