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Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles I: Fixed point algebra, simplicity and the tracial state space 由向量束扭转的部分自形的 Cuntz-Pimsner 代数 I：定点代数、简单性和三态空间
arXiv - MATH - Operator Algebras Pub Date : 2024-08-19 DOI: arxiv-2408.10047
Aaron Kettner
{"title":"Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles I: Fixed point algebra, simplicity and the tracial state space","authors":"Aaron Kettner","doi":"arxiv-2408.10047","DOIUrl":"https://doi.org/arxiv-2408.10047","url":null,"abstract":"We associate a $C^*$-algebra to a partial action of the integers acting on\u0000the base space of a vector bundle, using the framework of Cuntz--Pimsner\u0000algebras. We investigate the structure of the fixed point algebra under the\u0000canonical gauge action, and show that it arises from a continuous field of\u0000$C^*$-algebras over the base space, generalising results of Vasselli. We also\u0000analyse the ideal structure, and show that for a free action, ideals correspond\u0000to open invariant subspaces of the base space. This shows that if the action is\u0000free and minimal, then the Cuntz--Pimsner algebra is simple. Finally we\u0000establish a bijective corrrespondence between tracial states and invariant\u0000measures on the base space, thereby calculating part of the Elliott invariant.\u0000This generalizes results about the $C^*$-algebras associated to homeomorphisms\u0000twisted by vector bundles of Adamo, Archey, Forough, Georgescu, Jeong, Strung\u0000and Viola.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195053","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
Primitive Ideal Space of $C^*(R_+)rtimes R^times$ $C^*(R_+)rtimes R^times$ 的原始理想空间
arXiv - MATH - Operator Algebras Pub Date : 2024-08-19 DOI: arxiv-2408.09863
Xiaohui Chen, Hui Li
{"title":"Primitive Ideal Space of $C^*(R_+)rtimes R^times$","authors":"Xiaohui Chen, Hui Li","doi":"arxiv-2408.09863","DOIUrl":"https://doi.org/arxiv-2408.09863","url":null,"abstract":"For an integral domain $R$ satisfying certain condition, we characterize the\u0000primitive ideal space and its Jacobson topology of the semigroup crossed\u0000product $C^*(R_+) rtimes R^times$. The main example is when\u0000$R=mathbb{Z}[sqrt{-3}]$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195055","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
Topics in Algebra of Synchronous Games, Algebraic Graph Identities and Quantum NP-hardness Reductions 同步游戏代数、代数图同一性和量子 NP 难度降低专题
arXiv - MATH - Operator Algebras Pub Date : 2024-08-19 DOI: arxiv-2408.10114
Entong He
{"title":"Topics in Algebra of Synchronous Games, Algebraic Graph Identities and Quantum NP-hardness Reductions","authors":"Entong He","doi":"arxiv-2408.10114","DOIUrl":"https://doi.org/arxiv-2408.10114","url":null,"abstract":"We review the correspondence between a synchronous game and its associated\u0000game algebra. We slightly develop the work of Helton et al.[HMPS17] by\u0000proposing results on algebraic and locally commuting graph identities. Based on\u0000the theoretical works on noncommutative Nullstellens\"atze [BWHK23], we build\u0000computational tools involving Gr\"obner basis methods and semidefinite\u0000programming to check the existence of perfect strategies with specific models.\u0000We prove the equivalence between the hereditary and $C$-star models proposed in\u0000[HMPS17]. We also extend Ji's reduction $texttt{3-SAT}text{-star} leq_p\u0000texttt{3-Coloring}text{-star}$ [Ji13] and exhibit another instance of\u0000quantum-version NP-hardness reduction $texttt{3-SAT}text{-star} leq_p\u0000texttt{Clique}text{-star}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195057","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
K-theoretic invariants for unital Kirchberg algebras with finitely generated K-groups 具有有限生成的 K 群的单元基希贝格代数的 K 理论不变式
arXiv - MATH - Operator Algebras Pub Date : 2024-08-18 DOI: arxiv-2408.09359
Kengo Matsumoto, Taro Sogabe
{"title":"K-theoretic invariants for unital Kirchberg algebras with finitely generated K-groups","authors":"Kengo Matsumoto, Taro Sogabe","doi":"arxiv-2408.09359","DOIUrl":"https://doi.org/arxiv-2408.09359","url":null,"abstract":"We introduce a hierarchy for unital Kirchberg algebras with finitely\u0000generated K-groups by which the 1st and 2nd homotopy groups of the automorphism\u0000groups serve as a complete invariant of classification. We also give a complete\u0000invariant specific to the case of unital Kirchberg algebras with finitely\u0000generated K-groups and provide a useful tool to classify the Cuntz--Krieger\u0000algebras.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195056","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
Quantum paving: When sphere packings meet Gabor frames 量子铺路：当球体填料遇到 Gabor 框架
arXiv - MATH - Operator Algebras Pub Date : 2024-08-16 DOI: arxiv-2408.08975
Markus Faulhuber, Thomas Strohmer
{"title":"Quantum paving: When sphere packings meet Gabor frames","authors":"Markus Faulhuber, Thomas Strohmer","doi":"arxiv-2408.08975","DOIUrl":"https://doi.org/arxiv-2408.08975","url":null,"abstract":"We introduce the new problems of quantum packing, quantum covering, and\u0000quantum paving. These problems arise naturally when considering an algebra of\u0000non-commutative operators that is deeply rooted in quantum physics as well as\u0000in Gabor analysis. Quantum packing and quantum covering show similarities with\u0000energy minimization and the dual problem of polarization. Quantum paving, in\u0000turn, aims to simultaneously optimize both quantum packing and quantum\u0000covering. Classical sphere packing and covering hint the optimal configurations\u0000for our new problems. We present solutions in certain cases, state several\u0000conjectures related to quantum paving and discuss some applications.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195058","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
Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version 张量积的分割和中间因子定理：连续版本
arXiv - MATH - Operator Algebras Pub Date : 2024-08-16 DOI: arxiv-2408.08635
Tattwamasi Amrutam, Yongle Jiang
{"title":"Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version","authors":"Tattwamasi Amrutam, Yongle Jiang","doi":"arxiv-2408.08635","DOIUrl":"https://doi.org/arxiv-2408.08635","url":null,"abstract":"Let $G$ be a discrete group. Given unital $G$-$C^*$-algebras $mathcal{A}$\u0000and $mathcal{B}$, we give an abstract condition under which every\u0000$G$-subalgebra $mathcal{C}$ of the form $mathcal{A}subset mathcal{C}subset\u0000mathcal{A}otimes_{text{min}}mathcal{B}$ is a tensor product. This\u0000generalizes the well-known splitting results in the context of $C^*$-algebras\u0000by Zacharias and Zsido. As an application, we prove a topological version of\u0000the Intermediate Factor theorem. When a product group\u0000$G=Gamma_1timesGamma_2$ acts (by a product action) on the product of\u0000corresponding $Gamma_i$-boundaries $partialGamma_i$, using the abstract\u0000condition, we show that every intermediate subalgebra\u0000$C(X)subsetmathcal{C}subset C(X)otimes_{text{min}}C(partialGamma_1times\u0000partialGamma_2)$ is a tensor product (under some additional assumptions on\u0000$X$). This can be considered as a topological version of the Intermediate\u0000Factor theorem. We prove that our assumptions are necessary and cannot\u0000generally be relaxed. We also introduce the notion of a uniformly rigid action\u0000for $C^*$-algebras and use it to give various classes of inclusions\u0000$mathcal{A}subset mathcal{A}otimes_{text{min}}mathcal{B}$ for which every\u0000invariant intermediate algebra is a tensor product.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195065","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
Sharp bottom spectrum and scalar curvature rigidity 锐底谱和标量曲率刚度
arXiv - MATH - Operator Algebras Pub Date : 2024-08-15 DOI: arxiv-2408.08245
Jinmin Wang, Bo Zhu
{"title":"Sharp bottom spectrum and scalar curvature rigidity","authors":"Jinmin Wang, Bo Zhu","doi":"arxiv-2408.08245","DOIUrl":"https://doi.org/arxiv-2408.08245","url":null,"abstract":"We prove a sharp upper bound for the bottom spectrum of Laplacian on\u0000geometrically contractible manifolds with scalar curvature lower bound, and\u0000characterize the distribution of scalar curvature when equality holds.\u0000Moreover, we prove a scalar curvature rigidity theorem if the manifold is the\u0000universal cover of a closed hyperbolic manifold.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195177","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
A Shubin pseudodifferential calculus on asymptotically conic manifolds 渐近圆锥流形上的舒宾伪微分学
arXiv - MATH - Operator Algebras Pub Date : 2024-08-15 DOI: arxiv-2408.08169
Thomas Krainer
{"title":"A Shubin pseudodifferential calculus on asymptotically conic manifolds","authors":"Thomas Krainer","doi":"arxiv-2408.08169","DOIUrl":"https://doi.org/arxiv-2408.08169","url":null,"abstract":"We present a global pseudodifferential calculus on asymptotically conic\u0000manifolds that generalizes (anisotropic versions of) Shubin's classical global\u0000pseudodifferential calculus on Euclidean space to this class of noncompact\u0000manifolds. Fully elliptic operators are shown to be Fredholm in an associated\u0000scale of Sobolev spaces, and to have parametrices in the calculus.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195059","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
Self-Dual Cone Systems and Tensor Products 自双锥系统和张量产品
arXiv - MATH - Operator Algebras Pub Date : 2024-08-14 DOI: arxiv-2408.07389
Tim Netzer
{"title":"Self-Dual Cone Systems and Tensor Products","authors":"Tim Netzer","doi":"arxiv-2408.07389","DOIUrl":"https://doi.org/arxiv-2408.07389","url":null,"abstract":"We prove the existence of self-dual tensor products for finite-dimensional\u0000convex cones and operator systems. This is a consequence of a more general\u0000result: Every cone system, which is contained in its dual, can be enlarged to a\u0000self-dual cone system. Using the setup of cone systems, we further describe how\u0000all functorial tensor products of finite-dimensional cones and operator systems\u0000explicitly arise from the minimal and maximal tensor product.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195066","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 support of free convolutions 关于自由卷积的支持
arXiv - MATH - Operator Algebras Pub Date : 2024-08-13 DOI: arxiv-2408.06573
Serban Belinschi, Hari Bercovici, Ching-Wei Ho
{"title":"On the support of free convolutions","authors":"Serban Belinschi, Hari Bercovici, Ching-Wei Ho","doi":"arxiv-2408.06573","DOIUrl":"https://doi.org/arxiv-2408.06573","url":null,"abstract":"We extend to arbitrary measures results of Bao, Erd\"os, Schnelli, Moreillon,\u0000and Ji on the connectedness of the supports of additive convolutions of\u0000measures on mathbb{R} and of free multiplicative convolutions of measures on\u0000mathbb{R}_+. More precisely, the convolution of two measures with connected\u0000supports also has connected support. The result holds without any absolute\u0000continuity or bounded support hypotheses on the measures being convolved. We\u0000also show that the results of Moreillon and Schnelli concerning the number of\u0000components of the support of a free additive convolution hold for arbitrary\u0000measures with bounded supports. Finally, we provide an approach to the\u0000corresponding results in the case of free multiplicative convolutions of\u0000probability measures on the unit circle.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195308","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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