Jensen's inequality for conditional expectations

Jensen's trace inequality in several variables

Jensen's trace inequality is established for every multivariable, convex function and every trace or trace-like functional on a C*-algebra.

Jensen's operator inequality and its converses

We give a general formulation of Jensen's operator inequality for unital fields of positive linear mappings, and we consider different types of converse inequalities.

Jensen's Operator Inequality

We establish what we consider to be the definitive versions of Jensen's operator inequality and Jensen's trace inequality for functions defined on an interval. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions used in earlier versions of the theory. As a consequence, we no longer need to impose conditions on the interval of definition. We show how this relates to the pinching inequality of Davis, and how Jensen's trace inequlity generalizes to C*-algebras..

Jensen's inequality for separately convex noncommutative functions

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 measu…

Jensen's Inequality and majorization

Let $\mathcal{A}$ be a $C^*$-algebra and $ϕ:\cA\to L(H)$ be a positive unital map. Then, for a convex function $f:I\to \mathbb{R}$ defined on some open interval and a self-adjoint element $a\in \mathcal{A}$ whose spectrum lies in $I$, we obtain a Jensen's-type inequality $f(ϕ(a)) \leq ϕ(f(a))$ where $\le$ denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered i.e., operator convex, monotone convex and arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.

带等号条件的Jensen迹不等式与非交换Lamperti定理

Jensen' s trace inequality with equality condition and noncommutative Lamperti's theorem

该研究针对半有限von Neumann代数，建立带等号条件的Jensen迹不等式，解答相关问题，推导非交换Lamperti型不等式，刻画一类F-范数非交换Orlicz空间上的线性等距，得到对应非交换Lamperti定理。