关于Stone-Weierstrass定理

Survey on a quantum stochastic extension of Stone's Theorem

From Kümmerer's investigations on stationary Markov processes has emerged an operator algebraic definition of white noises which captures many examples from classical as well as from non-commutative probability. Within non-commutative $L^p$-spaces associated to a white noise, the role of (non-)commutative Lévy processes is played by additive cocycles for the white noise shift, and moreover, the notion for exponentials of classical Lévy processes is generalized by unitary cocycles. As a main result we report a bijective correspondence between additive and unitary cocycles for white noise shifts…

局部紧量子群的协变 Stone-von Neumann 定理

The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups

本文将 Stone-von Neumann 定理推广至 Hilbert 模块上的局部紧量子动力学系统，引入一对模表示以涵盖多种模型，并提出了识别强正则局部紧量子群的新准则。

拓扑量子理论中的复数、幺正变换的单参数和Stone定理

Complex Numbers, One-Parameter of Unitary Transformations and Stone's Theorem in Topos Quantum Theory

拓扑理论首先由伊沙姆和巴特菲尔德提出，然后由伊沙姆和多林提出，作为一种替代的数学结构，在其中制定物理理论。特别是，它已被用来重新制定标准的量子力学，以这样一种方式，一种新型的逻辑是用来表示命题。近年来，随着概率、群和群变换的引入，这一主题取得了相当大的进展。在本文中，我们将介绍一个候选的复量值对象，并分析它的关系，真正的量值对象。通过定义这两个对象的Grothendieck k-扩张，将它们转化为阿贝尔群，可以在拓扑中定义内部单参数群。然后，我们使用这个新的定义来构建拓扑模拟的石头定理。

The Modular Stone-von Neumann Theorem

In this paper, we use the tools of nonabelian duality to formulate and prove a far-reaching generalization of the Stone-von Neumann Theorem to modular representations of actions and coactions of locally compact groups on elementary $ C^{\ast} $-algebras. This greatly extends the Covariant Stone-von Neumann Theorem for Actions of Abelian Groups recently proven by L. Ismert and the second author. Our approach is based on a new result about Hilbert $ C^{\ast} $-modules that is simple to state yet is widely applicable and can be used to streamline many previous arguments, so it represents an impro…

The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on $ C^{\ast} $-Algebras of Compact Operators

In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),α) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hil…