##### Department of Mathematics,

University of California San Diego

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### Math 243 - Functional Analysis Seminar

## Ian Charlesworth

#### KU Leuven

## Recent work on free Stein dimension

##### Abstract:

Free information theory is largely concerned with the following question: given a tuple of non-commutative random variables, what regularity properties of the algebra they generate can be inferred from assumptions about their joint distribution? This can include von Neumann algebraic properties, such as factoriality or absence of Cartan subalgebras, and free probabilistic properties, such as a lack of non-commutative rational relations. After giving some background, I will talk on free Stein dimension, a quantity which measures the ease of defining derivations on a tuple of non-commutative variables which turns out to be a $*$-algebra invariant. I will mention some recent results on its theory, including behaviour in the presence of algebraic relations as well as under direct sum and amplification of algebras. I will also mention some recent attempts to adapt its utility from polynomial algebras to W*-algebras, and time permitting, some cases where explicit estimates can be found on the Stein dimension of generating tuples of von Neumann algebras. This project is joint work with Brent Nelson.

Host: David Jekel

### November 9, 2021

### 10:00 AM

### Please email djekel@ucsd.edu for Zoom details.

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