分离凸非交换函数的Jensen不等式
Jensen's inequality for separately convex noncommutative functions
摘要：
经典的Jensen不等式断言，如果$X$是一个紧凸集，$f:K\to \mathbb{R}$是一个凸函数，那么对于$K$上的任何概率测度$μ$，$f(\text{bar}(μ))\le \int f\;dμ$，其中$\text{bar}(μ)$是$μ$的重心。最近，Davidson和Kennedy证明了Jensen不等式的一个非交换（“nc”）版本，该不等式适用于nc凸函数，该函数取矩阵值，用ucp映射代替概率测度。在经典情形下，若$f$仅为分离凸函数，则对任意乘积测度的概率测度，$f$仍满足Jensen不等式.本文证明了一个非交换的Jensen不等式，它适用于在每个变量上分别为nc凸的函数。这个不等式适用于一类满足Fubini定理的非交换类似物的UCP映射。这类ucp映射包括任何由博卡定理建立的ucp映射的自由积，或者任何在Mackotkowski的自由概率意义下条件自由的ucp映射。作为对自由概率的应用，我们得到了条件自由ucp映射应用于自由半圆族的一些算子不等式。
英文摘要：
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 $μ$ on $K$, that $f(\text{bar}(μ))\le \int f\;dμ$, where $\text{bar}(μ)$ is the barycenter of $μ$. 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 Młotkowski. As an application to free probability, we obtain some operator inequalities for conditionally free ucp maps applied to free semicircular families.
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