In this note, we give criteria on noncommutative integral kernels ensuring that integral operators on quantum torus belong to Schatten classes. With the engagement of a noncommutative Schwartz' kernel theorem on the quantum torus, a specific test for Schatten class properties of bounded operators on the quantum torus is established.
{"title":"Schatten classes on noncommutative tori: Kernel conditions","authors":"M. Ruzhansky, K. Zeng","doi":"arxiv-2407.09715","DOIUrl":"https://doi.org/arxiv-2407.09715","url":null,"abstract":"In this note, we give criteria on noncommutative integral kernels ensuring\u0000that integral operators on quantum torus belong to Schatten classes. With the\u0000engagement of a noncommutative Schwartz' kernel theorem on the quantum torus, a\u0000specific test for Schatten class properties of bounded operators on the quantum\u0000torus is established.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The method of Feynman-Kac perturbation of quantum stochastic processes has a long pedigree, with the theory usually developed within the framework of processes on von Neumann algebras. In this work, the theory of operator spaces is 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 may seem numerous, we provide auxiliary results that enable this to be simplified in many of the cases which arise in practice. A wide variety of examples is provided by way of illustration.
{"title":"Feynman-Kac perturbation of $C^*$ quantum stochastic flows","authors":"Alexander C. R. Belton, Stephen J. Wills","doi":"arxiv-2407.06732","DOIUrl":"https://doi.org/arxiv-2407.06732","url":null,"abstract":"The method of Feynman-Kac perturbation of quantum stochastic processes has a\u0000long pedigree, with the theory usually developed within the framework of\u0000processes on von Neumann algebras. In this work, the theory of operator spaces\u0000is exploited to enable a broadening of the scope to flows on $C^*$ algebras.\u0000Although the hypotheses that need to be verified in the general setting may\u0000seem numerous, we provide auxiliary results that enable this to be simplified\u0000in many of the cases which arise in practice. A wide variety of examples is\u0000provided by way of illustration.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we prove maximal inequality and ergodic theorems for state preserving actions on von Neumann algebra by an amenable, locally compact, second countable group equipped with the metric satisfying the doubling condition. The key idea is to use Hardy-Littlewood maximal inequality, a version of the transference principle, and certain norm estimates of differences between ergodic averages and martingales.
{"title":"Maximal Inequality Associated to Doubling Condition for State Preserving Actions","authors":"Panchugopal Bikram, Diptesh Saha","doi":"arxiv-2407.05642","DOIUrl":"https://doi.org/arxiv-2407.05642","url":null,"abstract":"In this article, we prove maximal inequality and ergodic theorems for state\u0000preserving actions on von Neumann algebra by an amenable, locally compact,\u0000second countable group equipped with the metric satisfying the doubling\u0000condition. The key idea is to use Hardy-Littlewood maximal inequality, a\u0000version of the transference principle, and certain norm estimates of\u0000differences between ergodic averages and martingales.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is intended to present the basic properties of $KO$-theory for real $C^*$-algebras and to explain its relationship with complex $K$-theory and with $KR$- theory. Whenever possible we will rely upon proofs in printed literature, particularly the work of Karoubi, Wood, Schr"oder, and more recent work 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 some deeper features of $KO$-theory for operator algebras may provide powerful new tools there. Commutative real $C^*$-algebras NOT of the form $C^R(X)$ will play a 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$KR^0(X, tau ))$ 。
{"title":"Real $K$-Theory for $C^*$-Algebras: Just the Facts","authors":"Jeff Boersema, Claude Schochet","doi":"arxiv-2407.05880","DOIUrl":"https://doi.org/arxiv-2407.05880","url":null,"abstract":"This paper is intended to present the basic properties of $KO$-theory for\u0000real $C^*$-algebras and to explain its relationship with complex $K$-theory and\u0000with $KR$- theory. Whenever possible we will rely upon proofs in printed\u0000literature, particularly the work of Karoubi, Wood, Schr\"oder, and more recent\u0000work of Boersema and J. M. Rosenberg. In addition, we shall explain how\u0000$KO$-theory is related to the Ten-Fold Way in physics and point out how some\u0000deeper features of $KO$-theory for operator algebras may provide powerful new\u0000tools there. Commutative real $C^*$-algebras NOT of the form $C^R(X)$ will play\u0000a special role. We also will identify Atiyah's $KR^0(X, tau ))$ in terms of\u0000$KO_0$ of an associated real $C^*$-algebra.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pierre Fima, François Le Maître, Kunal Mukherjee, Issan Patri
The Mar'echal topology, also called the Effros-Mar'echal topology, is a natural topology one can put on the space of all von Neumann subalgebras of a given von Neumann algebra. It is a result of Mar'echal from 1973 that this topology is Polish as soon as the ambient algebra has separable predual, but the 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 the topologies one can put on the space of weak-$*$ closed subspaces of a dual space. We also indicate how Michael's selection theorem can be used as a step towards Mar'echal's theorem, and how it simplifies the proof of an important selection result of Haagerup and Winsl{o}w for the Mar'echal topology. As an application, we show that the space of finite von Neumann algebras is $mathbfPi^0_3$-complete.
{"title":"Michael's selection theorem and applications to the Maréchal topology","authors":"Pierre Fima, François Le Maître, Kunal Mukherjee, Issan Patri","doi":"arxiv-2407.05776","DOIUrl":"https://doi.org/arxiv-2407.05776","url":null,"abstract":"The Mar'echal topology, also called the Effros-Mar'echal topology, is a\u0000natural topology one can put on the space of all von Neumann subalgebras of a\u0000given von Neumann algebra. It is a result of Mar'echal from 1973 that this\u0000topology is Polish as soon as the ambient algebra has separable predual, but\u0000the sketch of proof in her research announcement appears to have a small gap.\u0000Our main goal in this paper is to fill this gap by a careful look at the\u0000topologies one can put on the space of weak-$*$ closed subspaces of a dual\u0000space. We also indicate how Michael's selection theorem can be used as a step\u0000towards Mar'echal's theorem, and how it simplifies the proof of an important\u0000selection result of Haagerup and Winsl{o}w for the Mar'echal topology. As an\u0000application, we show that the space of finite von Neumann algebras is\u0000$mathbfPi^0_3$-complete.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141577549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally, in the specialized case where the higher-rank graph consists of one vertex, we are able to use the K"unneth theorem to explicitly compute the homology groups of 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。
{"title":"On The Evans Chain Complex","authors":"S. Joseph Lippert","doi":"arxiv-2407.06065","DOIUrl":"https://doi.org/arxiv-2407.06065","url":null,"abstract":"We elaborate on the construction of the Evans chain complex for higher-rank\u0000graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of\u0000the differential maps. These block matrices are then used to identify a wide\u0000family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally,\u0000in the specialized case where the higher-rank graph consists of one vertex, we\u0000are able to use the K\"unneth theorem to explicitly compute the homology groups\u0000of the Evans chain complex.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The notion of almost elementariness for a locally compact Hausdorff '{e}tale groupoid $mathcal{G}$ with a compact unit space was introduced by the authors as a sufficient condition ensuring the reduced groupoid $C^*$-algebra $C^*_r(mathcal{G})$ is (tracially) $mathcal{Z}$-stable and thus classifiable under additional natural assumption. In this paper, we explore the converse direction and show that many groupoids in the literature serving as models for classifiable $C^*$-algebras are almost elementary. In particular, for a large class $mathcal{C}$ of Elliott invariants and a $C^*$-algebra $A$ with $operatorname{Ell}(A)in mathcal{C}$, we show that $A$ is classifiable if and only if $A$ possesses a minimal, effective, amenable, second countable, almost elementary groupoid model, which leads to a groupoid-theoretic characterization of classifiability of $C^*$-algebras with certain Elliott invariants. Moreover, we build a connection between almost elementariness and pure infiniteness for groupoids and study obstructions to obtaining a transformation groupoid model for the Jiang-Su algebra $mathcal{Z}$.
{"title":"Almost elementary groupoid models for $C^*$-algebras","authors":"Xin Ma, Jianchao Wu","doi":"arxiv-2407.05251","DOIUrl":"https://doi.org/arxiv-2407.05251","url":null,"abstract":"The notion of almost elementariness for a locally compact Hausdorff '{e}tale\u0000groupoid $mathcal{G}$ with a compact unit space was introduced by the authors\u0000as a sufficient condition ensuring the reduced groupoid $C^*$-algebra\u0000$C^*_r(mathcal{G})$ is (tracially) $mathcal{Z}$-stable and thus classifiable\u0000under additional natural assumption. In this paper, we explore the converse\u0000direction and show that many groupoids in the literature serving as models for\u0000classifiable $C^*$-algebras are almost elementary. In particular, for a large\u0000class $mathcal{C}$ of Elliott invariants and a $C^*$-algebra $A$ with\u0000$operatorname{Ell}(A)in mathcal{C}$, we show that $A$ is classifiable if and\u0000only if $A$ possesses a minimal, effective, amenable, second countable, almost\u0000elementary groupoid model, which leads to a groupoid-theoretic characterization\u0000of classifiability of $C^*$-algebras with certain Elliott invariants. Moreover,\u0000we build a connection between almost elementariness and pure infiniteness for\u0000groupoids and study obstructions to obtaining a transformation groupoid model\u0000for the Jiang-Su algebra $mathcal{Z}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work extends the Mond-Pecaric method to functions with multiple operators as arguments by providing arbitrarily close approximations of the original functions. Instead of using linear functions to establish lower and upper bounds for multivariate functions as in prior work, we apply sigmoid functions to achieve these bounds with any specified error threshold based on the multivariate function approximation method proposed by Cybenko. This approach allows us to derive fundamental inequalities for multivariate hypercomplex functions, leading to new inequalities based on ratio and difference kinds. For applications about these new derived inequalities for multivariate hypercomplex functions, we first introduce a new concept called W-boundedness for hypercomplex functions by applying ratio kind multivariate hypercomplex inequalities. W-boundedness generalizes R-boundedness for norm mappings with input from Banach space. Additionally, we develop an approximation theory for multivariate hypercomplex functions and establish bounds algebra, including operator bounds and tail bounds algebra for multivariate 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。
{"title":"Generalized Multivariate Hypercomplex Function Inequalities and Their Applications","authors":"Shih-Yu Chang","doi":"arxiv-2407.05062","DOIUrl":"https://doi.org/arxiv-2407.05062","url":null,"abstract":"This work extends the Mond-Pecaric method to functions with multiple\u0000operators as arguments by providing arbitrarily close approximations of the\u0000original functions. Instead of using linear functions to establish lower and\u0000upper bounds for multivariate functions as in prior work, we apply sigmoid\u0000functions to achieve these bounds with any specified error threshold based on\u0000the multivariate function approximation method proposed by Cybenko. This\u0000approach allows us to derive fundamental inequalities for multivariate\u0000hypercomplex functions, leading to new inequalities based on ratio and\u0000difference kinds. For applications about these new derived inequalities for\u0000multivariate hypercomplex functions, we first introduce a new concept called\u0000W-boundedness for hypercomplex functions by applying ratio kind multivariate\u0000hypercomplex inequalities. W-boundedness generalizes R-boundedness for norm\u0000mappings with input from Banach space. Additionally, we develop an\u0000approximation theory for multivariate hypercomplex functions and establish\u0000bounds algebra, including operator bounds and tail bounds algebra for\u0000multivariate random tensors.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tristan Bice, Lisa Orloff Clark, Ying-Fen Lin, Kathryn McCormick
We prove that twisted groupoid C*-algebras are characterised, up to isomorphism, by having Cartan semigroups, a natural generalisation of normaliser semigroups of Cartan subalgebras. This extends the classic Kumjian-Renault theory to general twisted 'etale groupoid C*-algebras, even non-reduced C*-algebras of non-effective groupoids.
{"title":"Cartan semigroups and twisted groupoid C*-algebras","authors":"Tristan Bice, Lisa Orloff Clark, Ying-Fen Lin, Kathryn McCormick","doi":"arxiv-2407.05024","DOIUrl":"https://doi.org/arxiv-2407.05024","url":null,"abstract":"We prove that twisted groupoid C*-algebras are characterised, up to\u0000isomorphism, by having Cartan semigroups, a natural generalisation of\u0000normaliser semigroups of Cartan subalgebras. This extends the classic\u0000Kumjian-Renault theory to general twisted 'etale groupoid C*-algebras, even\u0000non-reduced C*-algebras of non-effective groupoids.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we fully characterize maximal representations of a C*-correspondence. This strengthens several earlier results. We demonstrate the criterion with diverse examples. We also describe the noncommutative Choquet boundary and provide additional counterexamples to Arveson's hyperrigidity conjecture following the counterexample recently found by the author and Adam Dor-On.
{"title":"Maximality of correspondence representations","authors":"Boris Bilich","doi":"arxiv-2407.04278","DOIUrl":"https://doi.org/arxiv-2407.04278","url":null,"abstract":"In this paper, we fully characterize maximal representations of a\u0000C*-correspondence. This strengthens several earlier results. We demonstrate the\u0000criterion with diverse examples. We also describe the noncommutative Choquet\u0000boundary and provide additional counterexamples to Arveson's hyperrigidity\u0000conjecture following the counterexample recently found by the author and Adam\u0000Dor-On.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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