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Schatten classes on noncommutative tori: Kernel conditions 非交换环上的夏顿类：内核条件
arXiv - MATH - Operator Algebras Pub Date : 2024-07-12 DOI: arxiv-2407.09715
M. Ruzhansky, K. Zeng
{"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":"52 1","pages":""},"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}
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
arXiv - MATH - Operator Algebras Pub Date : 2024-07-09 DOI: arxiv-2407.06732
Alexander C. R. Belton, Stephen J. Wills
{"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":"25 1","pages":""},"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}
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
arXiv - MATH - Operator Algebras Pub Date : 2024-07-08 DOI: arxiv-2407.05642
Panchugopal Bikram, Diptesh Saha
{"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":"14 1","pages":""},"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}
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
arXiv - MATH - Operator Algebras Pub Date : 2024-07-08 DOI: arxiv-2407.05880
Jeff Boersema, Claude Schochet
{"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":"71 1","pages":""},"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}
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
arXiv - MATH - Operator Algebras Pub Date : 2024-07-08 DOI: arxiv-2407.05776
Pierre Fima, François Le Maître, Kunal Mukherjee, Issan Patri
{"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":"1 1","pages":""},"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}
引用次数: 0
On The Evans Chain Complex 埃文斯连锁综合体
arXiv - MATH - Operator Algebras Pub Date : 2024-07-08 DOI: arxiv-2407.06065
S. Joseph Lippert
{"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":"69 1","pages":""},"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}
引用次数: 0
Almost elementary groupoid models for $C^*$-algebras C^*$ 算法的几乎基本群模型
arXiv - MATH - Operator Algebras Pub Date : 2024-07-07 DOI: arxiv-2407.05251
Xin Ma, Jianchao Wu
{"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":"50 1","pages":""},"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}
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
arXiv - MATH - Operator Algebras Pub Date : 2024-07-06 DOI: arxiv-2407.05062
Shih-Yu Chang
{"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":"56 1","pages":""},"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}
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
arXiv - MATH - Operator Algebras Pub Date : 2024-07-06 DOI: arxiv-2407.05024
Tristan Bice, Lisa Orloff Clark, Ying-Fen Lin, Kathryn McCormick
{"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":"57 29 1","pages":""},"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}
引用次数: 0
Maximality of correspondence representations 对应表示的最大值
arXiv - MATH - Operator Algebras Pub Date : 2024-07-05 DOI: arxiv-2407.04278
Boris Bilich
{"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":"14 1","pages":""},"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}
引用次数: 0
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