标题： 冯·诺依曼代数中相对熵的单调性和双向博戈留波夫不等式 显示英文标题

标题： Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras

摘要： 本文一方面研究了Araki-Uhlmann相对熵的一些单调性性质，另一方面研究了KMS态的无界扰动理论，这有助于在一般的冯·诺依曼代数中证明双向Bogoliubov不等式。 在介绍了算子代数理论和Tomita-Takesaki模理论的相关背景之后，定义了相对熵泛函并研究了其基本性质。 特别是，提供了Uhlmann关于相对熵的重要单调性定理的完整详细证明。 然后将利用该定理推导出由形式为$V \varOmega, V \varPhi \in \mathcal{H}$的向量诱导的正常泛函的相对熵的一系列单调性不等式，其中$V \in \mathscr{B}(\mathcal{H})$是一个合适的变换。 随后，介绍了冯·诺依曼代数中的扰动理论，重点在于遵循Dereziński-Jakšić-Pillet框架的KMS态的无界扰动。 然后将使用这一数学工具将最近刚对量子力学系统证明的相对自由能的双向Bogoliubov不等式推广到任意冯·诺依曼代数。 显示英文摘要

摘要： This text studies, on the one hand, certain monotonicity properties of the Araki-Uhlmann relative entropy and, on the other hand, unbounded perturbation theory of KMS-states which facilitates a proof of the two-sided Bogoliubov inequality in general von Neumann algebras. After introducing the necessary background from the theory of operator algebras and Tomita-Takesaki modular theory, the relative entropy functional is defined and its basic properties are studied. In particular, a full and detailed proof of Uhlmann's important monotonicity theorem for the relative entropy is provided. This theorem will then be used to derive a number of monotonicity inequalities for the relative entropy of normal functionals induced by vectors of the form $V \varOmega, V \varPhi \in \mathcal{H}$, where $V \in \mathscr{B}(\mathcal{H})$ is a suitable transformation. After that, an introduction to perturbation theory in von Neumann algebras is given, with an emphasis on unbounded perturbations of KMS-states following the framework of Derezi\'{n}ski-Jak\v{s}i\'{c}-Pillet. This mathematical apparatus will then be used to extend the two-sided Bogoliubov inequality for the relative free energy, which was very recently proved for quantum-mechanical systems, to arbitrary von Neumann algebras.