Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929665881407488 |
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| author | Reible, Benedikt M. |
| author_facet | Reible, Benedikt M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04564 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras Reible, Benedikt M. Operator Algebras Mathematical Physics 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. |
| title | Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras |
| topic | Operator Algebras Mathematical Physics |
| url | https://arxiv.org/abs/2501.04564 |