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Symmetry Reduction of States I 状态的对称约简1
2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-10-06 DOI: 10.4171/jncg/534
Philipp Schmitt, Matthias Schötz
{"title":"Symmetry Reduction of States I","authors":"Philipp Schmitt, Matthias Schötz","doi":"10.4171/jncg/534","DOIUrl":"https://doi.org/10.4171/jncg/534","url":null,"abstract":"We develop a general theory of symmetry reduction of states on (possibly non-commutative) *-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra $g$. The key idea advocated for in this article is that the \"correct\" notion of positivity on a *-algebra $A$ is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares $a^* a$ with $a in A$, but can be a more general one that depends on the example at hand, like pointwise positivity on *-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on $A$ thus depends on this choice of positivity on $A$, and the notion of positivity on the reduced algebra $A_{red}$ should be such that states on $A_{red}$ are obtained as reductions of certain states on $A$. We discuss three examples in detail: Reduction of the *-algebra of smooth functions on a Poisson manifold $M$, which reproduces the coisotropic reduction of $M$; reduction of the Weyl algebra with respect to translation symmetry; and reduction of the polynomial algebra with respect to a U(1)-action.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Discrete quantum structures I: Quantum predicate logic 离散量子结构I:量子谓词逻辑
2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-10-04 DOI: 10.4171/jncg/531
Andre Kornell
{"title":"Discrete quantum structures I: Quantum predicate logic","authors":"Andre Kornell","doi":"10.4171/jncg/531","DOIUrl":"https://doi.org/10.4171/jncg/531","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"2012 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Pontrjagin duality on multiplicative gerbes 乘法gerbe上的Pontrjagin对偶性
2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-09-28 DOI: 10.4171/jncg/528
Jaider Blanco, Bernardo Uribe, Konrad Waldorf
{"title":"Pontrjagin duality on multiplicative gerbes","authors":"Jaider Blanco, Bernardo Uribe, Konrad Waldorf","doi":"10.4171/jncg/528","DOIUrl":"https://doi.org/10.4171/jncg/528","url":null,"abstract":"We use Segal–Mitchison’s cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and define their representations. For a specific choice of representation, we construct its category of endomorphisms, and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fiberwise Pontrjagin dual of the original one, and therefore we called the pair of multiplicative gerbes “Pontrjagin dual”. We show that Pontrjagin dual multiplicative gerbes have equivalent categories of representations. In addition, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact, Lie groups are provided.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135343459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Deformation of algebras associated with group cocycles 群环代数的变形
2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-09-18 DOI: 10.4171/jncg/522
Makoto Yamashita
{"title":"Deformation of algebras associated with group cocycles","authors":"Makoto Yamashita","doi":"10.4171/jncg/522","DOIUrl":"https://doi.org/10.4171/jncg/522","url":null,"abstract":"We study deformation of algebras with coaction symmetry of reduced algebras of discrete groups, where the deformation parameter is givenby a continuous family of group $2$-cocycles. When the group satisfies the Baum--Connes conjecture with coefficients, we obtain an isomorphism of K-groups of the deformed algebras. This extends both the $theta$-deformation of Rieffel on $mathbb{T}^n$-actions, and a recent result of Echterhoff, Lück, Phillips, and Walters on the K-groups on the twisted group algebras.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135149773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 14
Aspects of noncommutative geometry of Bunce–Deddens algebras Bunce-Deddens代数非交换几何的几个方面
2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-09-13 DOI: 10.4171/jncg/521
Slawomir Klimek, Matt McBride, J. Wilson Peoples
{"title":"Aspects of noncommutative geometry of Bunce–Deddens algebras","authors":"Slawomir Klimek, Matt McBride, J. Wilson Peoples","doi":"10.4171/jncg/521","DOIUrl":"https://doi.org/10.4171/jncg/521","url":null,"abstract":"We define and study smooth subalgebras of Bunce–Deddens C∗-algebras. We discuss various aspects of noncommutative geometry of Bunce–Deddens algebras including derivations on smooth subalgebras, as well as $K$-theory and $K$-homology.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134989643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mixed $q$-deformed Araki–Woods von Neumann algebras 混合$q$变形Araki–Woods-von Neumann代数
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-08-30 DOI: 10.4171/jncg/513
Panchugopal Bikram, Rahul Kumar R, Kunal Mukherjee
{"title":"Mixed $q$-deformed Araki–Woods von Neumann algebras","authors":"Panchugopal Bikram, Rahul Kumar R, Kunal Mukherjee","doi":"10.4171/jncg/513","DOIUrl":"https://doi.org/10.4171/jncg/513","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49428186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Measurability, spectral densities, and hypertraces in noncommutative geometry 非交换几何中的可测性、谱密度和超迹线
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-08-08 DOI: 10.4171/jncg/511
F. Cipriani, J. Sauvageot
{"title":"Measurability, spectral densities, and hypertraces in noncommutative geometry","authors":"F. Cipriani, J. Sauvageot","doi":"10.4171/jncg/511","DOIUrl":"https://doi.org/10.4171/jncg/511","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47325293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homomorphisms into simple $mathcal{Z}$-stable $C^*$-algebras, II 到简单$mathcal｛Z｝$-稳定$C^*$-代数的同态，II
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-07-14 DOI: 10.4171/jncg/490
Huaxin Lin, G. Gong, Z. Niu
{"title":"Homomorphisms into simple $mathcal{Z}$-stable $C^*$-algebras, II","authors":"Huaxin Lin, G. Gong, Z. Niu","doi":"10.4171/jncg/490","DOIUrl":"https://doi.org/10.4171/jncg/490","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43171405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal 动力系统的同调与K -理论2。具有完全不相连的横向的小空间
2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-05-18 DOI: 10.4171/jncg/494
Valerio Proietti, Makoto Yamashita
{"title":"Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal","authors":"Valerio Proietti, Makoto Yamashita","doi":"10.4171/jncg/494","DOIUrl":"https://doi.org/10.4171/jncg/494","url":null,"abstract":"We apply our previous work on the relation between groupoid homology and $K$-theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets and prove that the associated spectral sequence shows Putnam's stable homology groups on the second sheet. Moreover, this homology is in fact isomorphic to the groupoid homology of the unstable equivalence relation.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135767506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Symmetries of simple $Amathbb{T}$-algebras 简单$Amathbb｛T｝$代数的对称性
IF 0.9 2区 数学
Journal of Noncommutative Geometry Pub Date : 2023-05-06 DOI: 10.4171/jncg/492
Yuanhang Zhang
{"title":"Symmetries of simple $Amathbb{T}$-algebras","authors":"Yuanhang Zhang","doi":"10.4171/jncg/492","DOIUrl":"https://doi.org/10.4171/jncg/492","url":null,"abstract":"","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47904038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","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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