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Rigidity of free product von Neumann algebras
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 12 / December 2016
- Published online by Cambridge University Press:
- 17 November 2016, pp. 2461-2492
- Print publication:
- December 2016
-
- Article
- Export citation
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Let
$I$ be any nonempty set and let 
$(M_{i},\unicode[STIX]{x1D711}_{i})_{i\in I}$ be any family of nonamenable factors, endowed with arbitrary faithful normal states, that belong to a large class 
${\mathcal{C}}_{\text{anti}\text{-}\text{free}}$ 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 
$(M,\unicode[STIX]{x1D711})=\ast _{i\in I}(M_{i},\unicode[STIX]{x1D711}_{i})$ , 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.
