算子代数

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新提交 (展示 1 之 1 条目 )

[1] arXiv:2508.03731 [中文pdf, pdf, html, 其他]

标题： 重新审视强次可加性的算子扩展 显示英文标题

标题： Revisiting the operator extension of strong subadditivity

Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming

评论： 欢迎留言

主题： 算子代数 (math.OA) ; 数学物理 (math-ph) ; 量子物理 (quant-ph)

We give a new proof of the operator extension of the strong subadditivity of von Neumann entropy $\rho_{AB} \otimes \sigma_{C}^{-1} \leq \rho_{A} \otimes \sigma_{BC}^{-1}$ by identifying the mathematical structure behind it as Connes' theory of spatial derivatives. This immediately generalizes the inequality to arbitrary inclusions of von Neumann algebras. In the case of standard representations, it reduces to the monotonicity of the relative modular operator. 显示英文摘要

We give a new proof of the operator extension of the strong subadditivity of von Neumann entropy $\rho_{AB} \otimes \sigma_{C}^{-1} \leq \rho_{A} \otimes \sigma_{BC}^{-1}$ by identifying the mathematical structure behind it as Connes' theory of spatial derivatives. This immediately generalizes the inequality to arbitrary inclusions of von Neumann algebras. In the case of standard representations, it reduces to the monotonicity of the relative modular operator.

交叉提交 (展示 1 之 1 条目 )

[2] arXiv:2508.03779 (交叉列表自 math.FA) [中文pdf, pdf, html, 其他]

标题： 局部测度空间上局部希尔伯特空间的直接积分 显示英文标题

标题： Direct integral of locally Hilbert spaces over a locally measure space

Chaitanya J. Kulkarni, Santhosh Kumar Pamula

评论： 33页。arXiv管理员注释：与arXiv:2409.01200存在大量文本重叠

主题： 泛函分析 (math.FA) ; 算子代数 (math.OA)

在本工作中，我们通过将经典的测度空间概念推广为局部测度空间，引入了局部Hilbert空间的直接积分的概念。我们证明了在局部测度空间上的一族局部Hilbert空间的直接积分形成一个局部Hilbert空间。然后，我们定义了此类直接积分上的两个重要的局部有界算子子类，即可分解的局部有界算子和可对角化的局部有界算子。我们证明了这些子类中的每一个都形成一个局部冯诺依曼代数，并且特别地，可对角化算子的局部冯诺依曼代数是交换的。最后，我们证明了可对角化算子的局部冯诺依曼代数与可分解算子的局部冯诺依曼代数的换位子相一致。 显示英文摘要

In this work, we introduce the concept of the direct integral of locally Hilbert spaces by generalizing the classical notion of a measure space to that of a locally measure space. We establish that the direct integral of a family of locally Hilbert spaces over a locally measure space forms a locally Hilbert space. We then define two important subclasses of locally bounded operators on such direct integrals, namely decomposable locally bounded operators and diagonalizable locally bounded operators. We show that each of these subclasses forms a locally von Neumann algebra, and in particular, that the locally von Neumann algebra of diagonalizable operators is abelian. Finally, we prove that the locally von Neumann algebra of diagonalizable operators coincides with the commutant of the locally von Neumann algebra of decomposable operators.

替换提交 (展示 2 之 2 条目 )

[3] arXiv:2408.14221 (替换) [中文pdf, pdf, html, 其他]

标题： 大脑功能作为连接组的热平衡状态出现 显示英文标题

标题： Brain functions emerge as thermal equilibrium states of the connectome

Elkaïoum M. Moutuou, Habib Benali

评论： 这是将发表在《物理评论研究》上的论文最终版本。感谢匿名审稿人的有益意见

主题： 神经与认知 (q-bio.NC) ; 无序系统与神经网络 (cond-mat.dis-nn) ; 统计力学 (cond-mat.stat-mech) ; 算子代数 (math.OA) ; 量子物理 (quant-ph)

神经科学中的一个基本观点是，认知功能——如感知、学习、记忆和运动——是由大脑的结构组织所塑造和限制的。 尽管在绘制和分析结构连接组方面取得了显著进展，但将大脑的物理结构与其功能能力联系起来的原则仍然难以捉摸。 在此，我们引入了一个代数量子模型来弥合理论上的这一空白，为连接组与涌现的大脑功能之间的关系提供了新的见解，同时将结构数据与功能预测联系起来。 利用已充分映射的秀丽隐杆线虫解剖和突触外连接组，我们证明了大脑功能——定义为神经系统的功能网络——作为由底层有向多重图的图代数导出的代数量子系统的热平衡状态出现。 具体而言，这些由Kubo-Martin-Schwinger（KMS）形式主义表征的平衡状态揭示了单个神经元如何参与功能网络的形成。 我们的模型通过两个关键特征阐明了神经回路中的结构-功能关系：(1) 一个功能连接组，用于界定拓扑驱动的神经元相互作用；(2) 一种整合能力（IC）指数，用于量化神经元协调和调节多种信息流的有效性。 这两个特征共同提供了一个统计学和机制性的信息流解释，并揭示了连接组的网络拓扑如何预测认知和复杂行为。 显示英文摘要

A fundamental idea in neuroscience is that cognitive functions -- such as perception, learning, memory, and locomotion -- are shaped and constrained by the brain's structural organization. Despite significant progress in mapping and analyzing structural connectomes, the principles linking the brain's physical architecture to its functional capabilities remain elusive. Here, we introduce an algebraic quantum model to bridge this theoretical gap, offering new insights into the relationship between the connectome and emergent brain functions, while connecting structural data to functional predictions. Using the well-mapped C. elegans anatomical and extrasynaptic connectomes, we demonstrate that brain functions, defined as functional networks of a neural system, emerge as thermal equilibrium states of an algebraic quantum system derived from the graph algebra of the underlying directed multigraph. Specifically, these equilibrium states, characterized by the Kubo-Martin-Schwinger (KMS) formalism, reveal how individual neurons contribute to functional network formation. Our model illuminates the structure-function relationship in neural circuits through two key features: (1) a functional connectome that delineates topologically driven neuronal interactions and (2) an Integration Capacity (IC) index that quantifies how effectively neurons coordinate and modulate diverse information flows. Together, these features provide a statistical and mechanistic account of information flow and reveal how the network topology of the connectome predicts cognition and complex behaviors.

[4] arXiv:2508.03271 (替换) [中文pdf, pdf, html, 其他]

标题： 关于相对熵的奇异极限 显示英文标题

标题： On singular limits of relative entropies

Feng Xu

评论： 13页；更正了一处参考文献

主题： 数学物理 (math-ph) ; 算子代数 (math.OA)

在本文中，我们推广了一个关键结果，该结果将某些相对熵的奇异极限与共形网设置中的指标相关联，这一结果最近在共形场论背景下的相对熵数学理论中发挥了重要作用。 显示英文摘要

In this paper we generalize a key result relating singular limits of certain relative entropies with index in the setting of conformal nets, which has played an important role recently in the mathematical theory of relative entropies in the context of Conformal Field Theory.