##### Department of Mathematics,

University of California San Diego

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### Math 243 - Seminar in Operator Algebras

## Daniel Hoff

#### UCLA

## Unique factorization of ${\rm II}_1$ factors of groups measure equivalent to products of hyperbolic groups

##### Abstract:

A ${\rm II}_1$ factor $M$ is called prime if it cannot be decomposed as a tensor product of ${\rm II}_1$ subfactors. Naturally, if $M$ is not prime, one asks if $M$ can be uniquely factored as a tensor product of prime subfactors. The first result in this direction is due to Ozawa and Popa in 2003, who gave a large class of groups $\mathcal{C}$ such that for any $\Gamma_1, \dots, \Gamma_n \in \mathcal{C}$, the associated von Neumann algebra $L(\Gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, L(\Gamma_n)$ is uniquely factored in a strong sense. This talk will consider the case where $\Gamma$ is icc group that is measure equivalent to a product of non-elementary hyperbolic groups. In joint work with Daniel Drimbe and Adrian Ioana, we show that any such $\Gamma$ admits a unique decomposition $\Gamma = \Gamma_1 \times \Gamma_2 \times \cdots \times \Gamma_n$ such that $L(\Gamma) = L(\Gamma_1) \,\overline{\otimes}\, \cdots \,\overline{\otimes}\, L(\Gamma_n)$ is uniquely factored in sense of Ozawa and Popa. Using this, we provide the first examples of prime ${\rm II}_1$ factors arising from lattices in higher rank Lie groups.

Host: Adrian Ioana

### January 31, 2017

### 2:00 PM

### AP&M 7321

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