Jensen's inequality for separately convex noncommutative functions

Abstract

Classically, Jensen's Inequality asserts that if $X$ is a compact convex set, and $f:K\to \mathbb{R}$ is a convex function, then for any probability measure $\mu$ on $K$, that $f(\text{bar}(\mu))\le \int f\;d\mu$, where $\text{bar}(\mu)$ is the barycenter of $\mu$. Recently, Davidson and Kennedy proved a noncommutative ("nc") version of Jensen's inequality that applies to nc convex functions, which take matrix values, with probability measures replaced by ucp maps. In the classical case, if $f$ is only a separately convex function, then $f$ still satisfies the Jensen inequality for any probability measure which is a product measure. We prove a noncommutative Jensen inequality for functions which are separately nc convex in each variable. The inequality holds for a large class of ucp maps which satisfy a noncommutative analogue of Fubini's theorem. This class of ucp maps includes any free product of ucp maps built from Boca's theorem, or any ucp map which is conditionally free in the free-probabilistic sense of M{\l}otkowski. As an application to free probability, we obtain some operator inequalities for conditionally free ucp maps applied to free semicircular families.