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Schatten classes on noncommutative tori: Kernel conditions 非交换环上的夏顿类:内核条件
Pub Date : 2024-07-12 DOI: arxiv-2407.09715
M. Ruzhansky, K. Zeng
In this note, we give criteria on noncommutative integral kernels ensuringthat integral operators on quantum torus belong to Schatten classes. With theengagement of a noncommutative Schwartz' kernel theorem on the quantum torus, aspecific test for Schatten class properties of bounded operators on the quantumtorus is established.
在本论文中,我们给出了非交换积分核的标准,以确保量子环上的积分算子属于沙腾类。通过量子环上的非交换施瓦茨核定理,我们建立了量子环上有界算子的沙腾类性质的特定检验。
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引用次数: 0
Feynman-Kac perturbation of $C^*$ quantum stochastic flows C^*$量子随机流的费曼-卡克扰动
Pub Date : 2024-07-09 DOI: arxiv-2407.06732
Alexander C. R. Belton, Stephen J. Wills
The method of Feynman-Kac perturbation of quantum stochastic processes has along pedigree, with the theory usually developed within the framework ofprocesses on von Neumann algebras. In this work, the theory of operator spacesis exploited to enable a broadening of the scope to flows on $C^*$ algebras.Although the hypotheses that need to be verified in the general setting mayseem numerous, we provide auxiliary results that enable this to be simplifiedin many of the cases which arise in practice. A wide variety of examples isprovided by way of illustration.
量子随机过程的费曼-卡克扰动方法有着悠久的历史,其理论通常是在冯-诺依曼代数的过程框架内发展起来的。在这项研究中,我们利用算子空间的理论,将研究范围扩大到 $C^*$ 矩阵上的流动。虽然在一般情况下需要验证的假设似乎很多,但我们提供了一些辅助结果,使实际中出现的许多情况得以简化。为了说明问题,我们提供了各种各样的例子。
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引用次数: 0
Maximal Inequality Associated to Doubling Condition for State Preserving Actions 与状态保持行动的加倍条件相关的最大不等式
Pub Date : 2024-07-08 DOI: arxiv-2407.05642
Panchugopal Bikram, Diptesh Saha
In this article, we prove maximal inequality and ergodic theorems for statepreserving actions on von Neumann algebra by an amenable, locally compact,second countable group equipped with the metric satisfying the doublingcondition. The key idea is to use Hardy-Littlewood maximal inequality, aversion of the transference principle, and certain norm estimates ofdifferences between ergodic averages and martingales.
在这篇文章中,我们证明了冯-诺依曼代数上的保态作用的最大不等式和遍历定理,这些保态作用是由一个可亲的、局部紧凑的、第二可数的、配备了满足加倍条件的度量的群来实现的。其关键思路是利用哈代-利特尔伍德最大不等式、转移原理的反演以及对遍历平均数与马氏数之间差异的某些规范估计。
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引用次数: 0
Real $K$-Theory for $C^*$-Algebras: Just the Facts C^*$代数的实$K$理论:事实而已
Pub Date : 2024-07-08 DOI: arxiv-2407.05880
Jeff Boersema, Claude Schochet
This paper is intended to present the basic properties of $KO$-theory forreal $C^*$-algebras and to explain its relationship with complex $K$-theory andwith $KR$- theory. Whenever possible we will rely upon proofs in printedliterature, particularly the work of Karoubi, Wood, Schr"oder, and more recentwork of Boersema and J. M. Rosenberg. In addition, we shall explain how$KO$-theory is related to the Ten-Fold Way in physics and point out how somedeeper features of $KO$-theory for operator algebras may provide powerful newtools there. Commutative real $C^*$-algebras NOT of the form $C^R(X)$ will playa special role. We also will identify Atiyah's $KR^0(X, tau ))$ in terms of$KO_0$ of an associated real $C^*$-algebra.
本文旨在介绍实$C^*$格的$KO$理论的基本性质,并解释它与复$K$理论和$KR$理论的关系。在可能的情况下,我们将依靠印刷文献中的证明,特别是卡鲁比、伍德、施罗德的工作,以及波尔塞马和罗森伯格的最新工作。此外,我们还将解释 $KO$ 理论与物理学中的 "十重道 "是如何相关的,并指出算子代数的 $KO$ 理论的一些更深层次的特征是如何为物理学提供强大的新工具的。非$C^R(X)$形式的交换实$C^*$数组将发挥特殊作用。我们还将根据相关实$C^*$代数的$KO_0$来确定阿蒂亚的$KR^0(X, tau ))$ 。
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引用次数: 0
Michael's selection theorem and applications to the Maréchal topology 迈克尔选择定理及其在马雷夏尔拓扑学中的应用
Pub Date : 2024-07-08 DOI: arxiv-2407.05776
Pierre Fima, François Le Maître, Kunal Mukherjee, Issan Patri
The Mar'echal topology, also called the Effros-Mar'echal topology, is anatural topology one can put on the space of all von Neumann subalgebras of agiven von Neumann algebra. It is a result of Mar'echal from 1973 that thistopology is Polish as soon as the ambient algebra has separable predual, butthe sketch of proof in her research announcement appears to have a small gap.Our main goal in this paper is to fill this gap by a careful look at thetopologies one can put on the space of weak-$*$ closed subspaces of a dualspace. We also indicate how Michael's selection theorem can be used as a steptowards Mar'echal's theorem, and how it simplifies the proof of an importantselection result of Haagerup and Winsl{o}w for the Mar'echal topology. As anapplication, we show that the space of finite von Neumann algebras is$mathbfPi^0_3$-complete.
马歇尔拓扑学(Mar'echal topology),也叫埃夫罗斯-马歇尔拓扑学(Effros-Mar'echal topology),是一种可以放在给定冯-诺依曼代数的所有冯-诺依曼子代数空间上的自然拓扑学。本文的主要目标是通过仔细研究可以放在对偶空间的弱-$*$封闭子空间上的拓扑来填补这一空白。我们还指出迈克尔选择定理如何被用作迈向马歇尔定理的一步,以及它如何简化了哈格鲁普和温斯洛对马歇尔拓扑的一个重要选择结果的证明。作为应用,我们证明了有限冯诺伊曼代数空间是完整的。
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引用次数: 0
On The Evans Chain Complex 埃文斯连锁综合体
Pub Date : 2024-07-08 DOI: arxiv-2407.06065
S. Joseph Lippert
We elaborate on the construction of the Evans chain complex for higher-rankgraph $C^*$-algebras. Specifically, we introduce a block matrix presentation ofthe differential maps. These block matrices are then used to identify a widefamily of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally,in the specialized case where the higher-rank graph consists of one vertex, weare able to use the K"unneth theorem to explicitly compute the homology groupsof the Evans chain complex.
我们详细阐述了高阶图 $C^*$ 算法的埃文斯链复数构造。具体来说,我们引入了微分映射的分块矩阵表述。然后,这些分块矩阵被用来识别具有琐K理论的高阶图$C^*$数组。此外,在高阶图由一个顶点组成的特殊情况下,我们能够使用 K ("unneth")定理来明确计算埃文斯链复数的同调群。
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引用次数: 0
Almost elementary groupoid models for $C^*$-algebras C^*$ 算法的几乎基本群模型
Pub Date : 2024-07-07 DOI: arxiv-2407.05251
Xin Ma, Jianchao Wu
The notion of almost elementariness for a locally compact Hausdorff '{e}talegroupoid $mathcal{G}$ with a compact unit space was introduced by the authorsas a sufficient condition ensuring the reduced groupoid $C^*$-algebra$C^*_r(mathcal{G})$ is (tracially) $mathcal{Z}$-stable and thus classifiableunder additional natural assumption. In this paper, we explore the conversedirection and show that many groupoids in the literature serving as models forclassifiable $C^*$-algebras are almost elementary. In particular, for a largeclass $mathcal{C}$ of Elliott invariants and a $C^*$-algebra $A$ with$operatorname{Ell}(A)in mathcal{C}$, we show that $A$ is classifiable if andonly if $A$ possesses a minimal, effective, amenable, second countable, almostelementary groupoid model, which leads to a groupoid-theoretic characterizationof classifiability of $C^*$-algebras with certain Elliott invariants. Moreover,we build a connection between almost elementariness and pure infiniteness forgroupoids and study obstructions to obtaining a transformation groupoid modelfor the Jiang-Su algebra $mathcal{Z}$.
作者提出了一个概念,即对于具有紧凑单位空间的局部紧凑 Hausdorff '{e}talegroupoid $mathcal{G}$ 来说,几乎元素性是一个充分条件,可以确保还原的基元 $C^*$-algebra$C^*_r(mathcal{G})$ 是(tracially)$mathcal{Z}$ 稳定的,从而在额外的自然假设下是可分类的。在本文中,我们探索了对话方向,并证明了文献中许多作为可分类 $C^*$ 算法模型的基元几乎都是基本的。特别是,对于埃利奥特不变式的大类 $mathcal{C}$ 和在 mathcal{C}$ 中具有$operatorname{Ell}(A)的$C^*$-代数 $A$,我们证明,如果且只有当 $A$ 具有一个最小值时,$A$ 才是可分类的、我们证明,只有当且仅当 $A$ 拥有一个最小的、有效的、友好的、第二可数的、几乎是元素的类群模型时,$A$ 才是可分类的。此外,我们还在几乎元元性和纯无限性之间建立了联系,并研究了江苏代数 $mathcal{Z}$ 获得变换群元模型的障碍。
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引用次数: 0
Generalized Multivariate Hypercomplex Function Inequalities and Their Applications 广义多元超复变函数不等式及其应用
Pub Date : 2024-07-06 DOI: arxiv-2407.05062
Shih-Yu Chang
This work extends the Mond-Pecaric method to functions with multipleoperators as arguments by providing arbitrarily close approximations of theoriginal functions. Instead of using linear functions to establish lower andupper bounds for multivariate functions as in prior work, we apply sigmoidfunctions to achieve these bounds with any specified error threshold based onthe multivariate function approximation method proposed by Cybenko. Thisapproach allows us to derive fundamental inequalities for multivariatehypercomplex functions, leading to new inequalities based on ratio anddifference kinds. For applications about these new derived inequalities formultivariate hypercomplex functions, we first introduce a new concept calledW-boundedness for hypercomplex functions by applying ratio kind multivariatehypercomplex inequalities. W-boundedness generalizes R-boundedness for normmappings with input from Banach space. Additionally, we develop anapproximation theory for multivariate hypercomplex functions and establishbounds algebra, including operator bounds and tail bounds algebra formultivariate random tensors.
这项工作通过提供原始函数的任意近似值,将蒙德-佩卡里克方法扩展到了以多个运算符为参数的函数。我们不再像以前的工作那样使用线性函数来建立多元函数的上下限,而是根据 Cybenko 提出的多元函数逼近方法,使用 sigmoid 函数来实现这些具有任意指定误差阈值的上下限。通过这种方法,我们可以推导出多元超复杂函数的基本不等式,从而得出基于比率和差分类型的新不等式。关于这些新导出不等式在多元超复变函数中的应用,我们首先通过应用比类多元超复变不等式,引入了一个新概念,即超复变函数的 W 有界性。W 有界性概括了从巴拿赫空间输入的规范映射的 R 有界性。此外,我们还发展了多元超复函数的近似理论,并建立了边界代数,包括多元随机张量形式的算子边界和尾边界代数。
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引用次数: 0
Cartan semigroups and twisted groupoid C*-algebras 笛卡尔半群和扭曲类群 C* 结构
Pub Date : 2024-07-06 DOI: arxiv-2407.05024
Tristan Bice, Lisa Orloff Clark, Ying-Fen Lin, Kathryn McCormick
We prove that twisted groupoid C*-algebras are characterised, up toisomorphism, by having Cartan semigroups, a natural generalisation ofnormaliser semigroups of Cartan subalgebras. This extends the classicKumjian-Renault theory to general twisted 'etale groupoid C*-algebras, evennon-reduced C*-algebras of non-effective groupoids.
我们证明了扭曲类群 C*-藻类的特征是具有 Cartan 半群(Cartan 子藻类的归一化半群的自然概括),直至同构。这就把经典的库姆江-雷诺理论扩展到了一般的扭转类群 C*-数,甚至是非有效类群的非还原 C*-数。
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引用次数: 0
Maximality of correspondence representations 对应表示的最大值
Pub Date : 2024-07-05 DOI: arxiv-2407.04278
Boris Bilich
In this paper, we fully characterize maximal representations of aC*-correspondence. This strengthens several earlier results. We demonstrate thecriterion with diverse examples. We also describe the noncommutative Choquetboundary and provide additional counterexamples to Arveson's hyperrigidityconjecture following the counterexample recently found by the author and AdamDor-On.
在本文中,我们完全描述了 C* 对应的最大表示。这加强了之前的几个结果。我们用不同的例子证明了这一标准。我们还描述了非交换卓奎特边界(non-commutative Choquetboundary),并在作者和亚当-多昂(AdamDor-On)最近发现的反例之后,为阿维森的超刚性猜想提供了更多反例。
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arXiv - MATH - Operator Algebras
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