##### Department of Mathematics,

University of California San Diego

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### Operator Algebras Seminar

## Cyril Houdayer

#### Universite Paris-Est Marne-la-Vallee

## Asymptotic structure and rigidity of free product von Neumann algebras

##### Abstract:

I will give an overview of recent results obtained in joint work with Yoshimichi Ueda on the structure and the rigidity of arbitrary free product von Neumann algebras. First, I will explain that in any free product von Neumann algebra, any amenable von Neumann subalgebra that has a diffuse intersection with one of the free components is necessarily contained in this free component. This result completely settles the problem of maximal amenability inside free product von Neumann algebras. Then I will present new Kurosh-type rigidity results for free product von Neumann algebras. Namely, I will explain that for any family of nonamenable factors belonging to a large class of (possibly type III) factors including nonprime factors, nonfull factors and factors with a Cartan subalgebra, the corresponding free product von Neumann algebra with respect to arbitrary states retains the cardinality of the family as well as each factor up to unitary conjugacy, after permutation of the indices.

Host: Adrian Ioana

### June 8, 2015

### 3:00 PM

### AP&M 5829

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