von Neumann代数中相对熵的单调性与双侧Bogoliubov不等式
Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras
AI总结：
本文证明Araki-Uhlmann相对熵的Uhlmann单调性定理并推导相关不等式，同时利用KMS态无界扰动理论将相对自由能的双侧Bogoliubov不等式推广至任意von Neumann代数。
摘要：
本文一方面研究Araki-Uhlmann相对熵的某些单调性，另一方面研究KMS态的无界扰动理论，该理论有助于在一般von Neumann代数中证明双侧Bogoliubov不等式。在介绍算子代数理论和Tomita-Takesaki模理论所需的背景之后，本文定义了相对熵泛函并研究其基本性质。特别地，本文给出了Uhlmann关于相对熵的重要单调性定理的完整且详细的证明。随后，该定理被用于推导由形如$V \varOmega, V \varPhi \in \mathcal{H}$的向量诱导的正规泛函的相对熵的若干单调性不等式，其中$V \in \mathscr{B}(\mathcal{H})$是适当的变换。之后，本文介绍了von Neumann代数中的扰动理论，重点遵循Dereziński-Jakšić-Pillet框架讨论KMS态的无界扰动。接着，这一数学工具被用于将相对自由能的双侧Bogoliubov不等式推广到任意von Neumann代数；该不等式最近刚在量子力学系统中得到证明。
英文摘要：
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 Dereziński-Jakšić-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.
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