Jensen's Inequality for separately convex noncommutative functions

Adam Humeniuk (University of Waterloo)

Thursday 10th March, 2022 16:00-17:00 Maths 311B


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 μ is a probability measure, then ∫ f dμ is at least as big as the evaluation of f at the "average point" or barycenter of μ. The noncommutative version of Jensen's Inequality holds for convex noncommutative functions with μ replaced by any ucp map. I'll show how the much broader class of multivariable noncommutative 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.

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