Department of Mathematics

Operator Algebras Reading

  •  Brent Nelson, MSU
  •  Free transport, II
  •  11/14/2022
  •  4:00 PM - 5:30 PM
  •  C517 Wells Hall
  •  Brent Nelson (banelson@msu.edu)

In operator algebras, specifically free probability, free transport is a technique for producing state-preserving isomorphisms between C* and von Neumann algebras that was developed by Guionnet and Shlyakhtenko in their 2014 Inventiones paper. The inspiration for their work comes from the field of optimal transport, specifically work of Brenier from 1991 who showed that under very mild assumptions one can push forward a probability measure on $\mathbb{R}^n$ to the Gaussian measure. In the non-commutative case, Guionnet and Shlyakhtenko showed that if $x_1,\ldots, x_n$ are self-adjoint operators in a tracial von Neumann algebra $(M,\tau)$ whose distribution satisfies an "integration-by-parts" formula up to a small perturbation, then these operators generate a copy of the free group factor $L(\mathbb{F}_n)$. In this series of talks, I will give an overview of their proof, discuss some applications of their result, and survey the current state of free transport theory.

 

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Michigan State University
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