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.
{"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}
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.
{"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}
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.
研究了离散群约化代数中具有共作用对称的代数的变形，其中变形参数由群$2$-环的连续族给出。当群满足带系数的Baum—Connes猜想时，我们得到了变形代数k群的一个同构。这扩展了Rieffel在$mathbb{T}^n$-作用上的$theta$-变形，以及Echterhoff, l k, Phillips和Walters关于扭曲群代数上k -群的最新结果。
{"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}
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.
定义并研究了Bunce-Deddens C * -代数的光滑子代数。讨论了Bunce-Deddens代数非交换几何的各个方面，包括光滑子代数上的导数，以及$K$-理论和$K$-同调。
{"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}
{"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}
{"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}
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.
{"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}
{"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}
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