We associate a $C^*$-algebra to a partial action of the integers acting on the base space of a vector bundle, using the framework of Cuntz--Pimsner algebras. We investigate the structure of the fixed point algebra under the canonical gauge action, and show that it arises from a continuous field of $C^*$-algebras over the base space, generalising results of Vasselli. We also analyse the ideal structure, and show that for a free action, ideals correspond to open invariant subspaces of the base space. This shows that if the action is free and minimal, then the Cuntz--Pimsner algebra is simple. Finally we establish a bijective corrrespondence between tracial states and invariant measures on the base space, thereby calculating part of the Elliott invariant. This generalizes results about the $C^*$-algebras associated to homeomorphisms twisted by vector bundles of Adamo, Archey, Forough, Georgescu, Jeong, Strung and Viola.
{"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":"108 1","pages":""},"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}
For an integral domain $R$ satisfying certain condition, we characterize the primitive ideal space and its Jacobson topology of the semigroup crossed product $C^*(R_+) rtimes R^times$. The main example is when $R=mathbb{Z}[sqrt{-3}]$.
{"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":"2 1","pages":""},"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}
We review the correspondence between a synchronous game and its associated game algebra. We slightly develop the work of Helton et al.[HMPS17] by proposing results on algebraic and locally commuting graph identities. Based on the theoretical works on noncommutative Nullstellens"atze [BWHK23], we build computational tools involving Gr"obner basis methods and semidefinite programming to check the existence of perfect strategies with specific models. We prove the equivalence between the hereditary and $C$-star models proposed in [HMPS17]. We also extend Ji's reduction $texttt{3-SAT}text{-star} leq_p texttt{3-Coloring}text{-star}$ [Ji13] and exhibit another instance of quantum-version NP-hardness reduction $texttt{3-SAT}text{-star} leq_p texttt{Clique}text{-star}$.
{"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":"23 1","pages":""},"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}
We introduce a hierarchy for unital Kirchberg algebras with finitely generated K-groups by which the 1st and 2nd homotopy groups of the automorphism groups serve as a complete invariant of classification. We also give a complete invariant specific to the case of unital Kirchberg algebras with finitely generated K-groups and provide a useful tool to classify the Cuntz--Krieger algebras.
我们为具有有限生成 K 群的基希贝格单原子代数引入了一种层次结构，通过这种层次结构，自变群的第 1 和第 2 同调群可作为分类的完全不变式。我们还给出了具有有限生成 K 群的单元基希贝格数组的完整不变量，并为 Cuntz--Kriegeralgebras 的分类提供了有用的工具。
{"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":"26 1","pages":""},"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}
We introduce the new problems of quantum packing, quantum covering, and quantum paving. These problems arise naturally when considering an algebra of non-commutative operators that is deeply rooted in quantum physics as well as in Gabor analysis. Quantum packing and quantum covering show similarities with energy minimization and the dual problem of polarization. Quantum paving, in turn, aims to simultaneously optimize both quantum packing and quantum covering. Classical sphere packing and covering hint the optimal configurations for our new problems. We present solutions in certain cases, state several conjectures related to quantum paving and discuss some applications.
{"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":"27 1","pages":""},"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}
Let $G$ be a discrete group. Given unital $G$-$C^*$-algebras $mathcal{A}$ and $mathcal{B}$, we give an abstract condition under which every $G$-subalgebra $mathcal{C}$ of the form $mathcal{A}subset mathcal{C}subset mathcal{A}otimes_{text{min}}mathcal{B}$ is a tensor product. This generalizes the well-known splitting results in the context of $C^*$-algebras by Zacharias and Zsido. As an application, we prove a topological version of the Intermediate Factor theorem. When a product group $G=Gamma_1timesGamma_2$ acts (by a product action) on the product of corresponding $Gamma_i$-boundaries $partialGamma_i$, using the abstract condition, we show that every intermediate subalgebra $C(X)subsetmathcal{C}subset C(X)otimes_{text{min}}C(partialGamma_1times partialGamma_2)$ is a tensor product (under some additional assumptions on $X$). This can be considered as a topological version of the Intermediate Factor theorem. We prove that our assumptions are necessary and cannot generally be relaxed. We also introduce the notion of a uniformly rigid action for $C^*$-algebras and use it to give various classes of inclusions $mathcal{A}subset mathcal{A}otimes_{text{min}}mathcal{B}$ for which every invariant intermediate algebra is a tensor product.
{"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":"3 1","pages":""},"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}
We prove a sharp upper bound for the bottom spectrum of Laplacian on geometrically contractible manifolds with scalar curvature lower bound, and characterize the distribution of scalar curvature when equality holds. Moreover, we prove a scalar curvature rigidity theorem if the manifold is the universal cover of a closed hyperbolic manifold.
{"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":"168 1","pages":""},"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}
We present a global pseudodifferential calculus on asymptotically conic manifolds that generalizes (anisotropic versions of) Shubin's classical global pseudodifferential calculus on Euclidean space to this class of noncompact manifolds. Fully elliptic operators are shown to be Fredholm in an associated scale of Sobolev spaces, and to have parametrices in the calculus.
{"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":"2 1","pages":""},"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}
We prove the existence of self-dual tensor products for finite-dimensional convex cones and operator systems. This is a consequence of a more general result: Every cone system, which is contained in its dual, can be enlarged to a self-dual cone system. Using the setup of cone systems, we further describe how all functorial tensor products of finite-dimensional cones and operator systems explicitly arise from the minimal and maximal tensor product.
{"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":"10 1","pages":""},"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}
We extend to arbitrary measures results of Bao, Erd"os, Schnelli, Moreillon, and Ji on the connectedness of the supports of additive convolutions of measures on mathbb{R} and of free multiplicative convolutions of measures on mathbb{R}_+. More precisely, the convolution of two measures with connected supports also has connected support. The result holds without any absolute continuity or bounded support hypotheses on the measures being convolved. We also show that the results of Moreillon and Schnelli concerning the number of components of the support of a free additive convolution hold for arbitrary measures with bounded supports. Finally, we provide an approach to the corresponding results in the case of free multiplicative convolutions of probability measures on the unit circle.
我们将 Bao、Erd"os、Schnelli、Moreillon 和 Ji 关于 mathbb{R} 上度量的加法卷积和 mathbb{R}_+ 上度量的自由乘法卷积的支撑的连通性的结果推广到任意度量。更精确地说，两个具有连通支持的度量的卷积也具有连通支持。这一结果不需要对被卷积的度量作任何绝对连续或有界支持的假设即可成立。我们还证明了莫里永和施内利关于自由加法卷积的支持分量数的结果，对于有界支持的任意度量也成立。最后，我们提供了在单位圆上概率度量的自由乘法卷积情况下获得相应结果的方法。
{"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":"86 1","pages":""},"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}
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