##### Department of Mathematics,

University of California San Diego

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### Math 243 - Functional Analysis Seminar

## David Jekel

#### UC San Diego

## Non-commutative smooth functions and non-commutative probability distributions

##### Abstract:

In free probability theory, there is no direct analog of density for probability distributions, but there is something like a notion of ``log-density'' in the study of free Gibbs laws and free score functions. Non-commutative notions of smoothness are important for studying both these log-densities and the functions used for changes of variables (or transport of measure) in free probability. In the single-variable setting, we have a good understanding of the smoothness properties of a function $f: \mathbb{R} \to \mathbb{R}$ applied to self-adjoint operators thanks to the work of Peller, Aleksandrov, and Nazarov; see the recent paper of Evangelos Nikitopoulos. However, in the multivariable setting, much of the literature has used classes of functions that are either too restrictive (such as analytic functions) or very technical to define (such as Dabrowski, Guionnet, and Shlyakhtenko's smooth functions where Haagerup tensor norms were used for the derivatives). We will discuss a notion of tracial non-commutative smooth functions that is modeled on trace polynomials. These smooth functions have many desirable properties, such as a chain rule, good behavior under conditional expectations, and a natural way to incorporate the one-variable functional calculus. We will sketch current work about the smooth transport of measure for free Gibbs laws as well as future directions in relating these smooth functions to free SDE.

### November 17, 2020

### 10:00 AM

### Contact mtwiersma@ucsd.edu for zoom info

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