Jensen's trace inequality in several variables
英文摘要:
Jensen's trace inequality is established for every multivariable, convex function and every trace or trace-like functional on a C*-algebra.
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Frank Hansen, Gert K. Pedersen
Jensen's trace inequality is established for every multivariable, convex function and every trace or trace-like functional on a C*-algebra.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
We study conditional expectations generated by an abelian $ C^* $-subalgebra in the centralizer of a positive functional. We formulate and prove Jensen's inequality for functions of several variables with respect to this type of conditional expectations, and we obtain as a corollary Jensen's inequality for expectation values.
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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..
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…
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.
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Jensen's inequality for partial traces in von Neumann algebras
本文在半有限冯·诺依曼代数中证明了部分追踪的詹森不等式,并扩展至一般冯·诺依曼代数框架。
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本文研究了从S-分歧到量子Jensen分歧的度量性质,扩展了S-分歧到迹C*-代数和有限von Neumann代数,并证明其平方根在特定框架下的度量性,同时回答了关于Hilbertianity的问题并推广了相关结果。
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Generalized Converses of Operator Jensens Inequalities with Applications to Hypercomplex Function Approximations and Bounds Algebra
本研究推广了Mond-Pecaric(MP)算子不等式方法,放宽凸函数与线性映射限制,推导超复函数新不等式,应用于函数逼近与界代数构建。
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Sharp estimates of quantum covering problems via a novel trace inequality
本文提出一种新的迹不等式,并利用它去除维度因子,从而尖锐化多种量子覆盖问题的单发可达界,且该界可推广至无限维空间。
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