# Analysis Student Seminar

Wednesday, August 4, 2021 — 3:00 PM EDT

Adam Humeniuk, Department of Pure Mathematics, University of Waterloo

"Jensen's Inequality for separately convex noncommutative functions"

I'll give a brief introduction to noncommutative (AKA "matrix") convexity, and some noncommutative analogues of classical convexity theorems. For instance, Jensen's Inequality asserts that if $f$ is a convex function and $\mu$ is a probability measure, then $\int f\; d\mu$ is at least as big as the evaluation of $f$ at the "average point" or barycenter of $\mu$. The noncommutative version of Jensen's Inequality holds for convex noncommutative ("nc") functions with $\mu$ replaced by any ucp map. I'll show how the much broader class of multivariable nc functions which are convex in each variable separately satisfy a Jensen Inequality for any "free product" of ucp maps. Such ucp maps show up naturally in free probability, and as a neat application we get some operator inequalities for free semicircular systems. In fact, Jensen's inequality holds for a bigger class of ucp maps which satisfy a sort of "nc Fubini theorem", but I don't know how to characterize these nicely. I'll give a minimal example that illustrates the difficulty, and suggest a connection to free probability.

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