Completely positive and completely bounded mutlipliers on rigid $C^{ast}$-tensor categories were introduced by Popa and Vaes. Using these notions, we define and study the Fourier-Stieltjes algebra, the Fourier algebra and the algebra of completely bounded multipliers of a rigid $C^{ast}$-tensor category. The rich structure that these algebras have in the setting of locally compact groups is still present in the setting of rigid $C^{ast}$-tensor categories. We also prove that Leptin's characterization of amenability still holds in this setting, and we collect some natural observations on property (T).
{"title":"The Fourier algebra of a rigid $C^{ast}$-tensor category","authors":"Yuki Arano, T. D. Laat, J. Wahl","doi":"10.4171/PRIMS/54-2-6","DOIUrl":"https://doi.org/10.4171/PRIMS/54-2-6","url":null,"abstract":"Completely positive and completely bounded mutlipliers on rigid $C^{ast}$-tensor categories were introduced by Popa and Vaes. Using these notions, we define and study the Fourier-Stieltjes algebra, the Fourier algebra and the algebra of completely bounded multipliers of a rigid $C^{ast}$-tensor category. The rich structure that these algebras have in the setting of locally compact groups is still present in the setting of rigid $C^{ast}$-tensor categories. We also prove that Leptin's characterization of amenability still holds in this setting, and we collect some natural observations on property (T).","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127056165","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}
Recently, W. Slofstra proved that the set of quantum correlations is not closed. We prove that the set of synchronous quantum correlations is not closed, which implies his result, by giving an example of a synchronous game that has a perfect quantum approximate strategy but no perfect quantum strategy. We also exhibit a graph for which the quantum independence number and the quantum approximate independence number are different. We prove new characterisations of synchronous quantum approximate correlations and synchronous quantum spatial correlations. We solve the synchronous approximation problem of Dykema and the second author, which yields a new equivalence of Connes' embedding problem in terms of synchronous correlations.
{"title":"A synchronous game for binary constraint systems","authors":"Se-Jin Kim, V. Paulsen, Christopher Schafhauser","doi":"10.1063/1.4996867","DOIUrl":"https://doi.org/10.1063/1.4996867","url":null,"abstract":"Recently, W. Slofstra proved that the set of quantum correlations is not closed. We prove that the set of synchronous quantum correlations is not closed, which implies his result, by giving an example of a synchronous game that has a perfect quantum approximate strategy but no perfect quantum strategy. We also exhibit a graph for which the quantum independence number and the quantum approximate independence number are different. We prove new characterisations of synchronous quantum approximate correlations and synchronous quantum spatial correlations. We solve the synchronous approximation problem of Dykema and the second author, which yields a new equivalence of Connes' embedding problem in terms of synchronous correlations.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123711833","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 show that an ergodic measure preserving action $Gamma curvearrowright (X,mu)$ of a discrete group $Gamma$ on a $sigma$-finite measure space $(X,mu)$ satisfies the local spectral gap property (introduced by Boutonnet, Ioana and Salehi Golsefidy) if and only if it is strongly ergodic. In fact, we prove a more general local spectral gap criterion in arbitrary von Neumann algebras. Using this criterion, we also obtain a short and elementary proof of Connes' spectral gap theorem for full $mathrm{II}_1$ factors as well as its recent generalization to full type $mathrm{III}$ factors.
{"title":"Strongly ergodic actions have local spectral gap","authors":"A. Marrakchi","doi":"10.1090/PROC/14034","DOIUrl":"https://doi.org/10.1090/PROC/14034","url":null,"abstract":"We show that an ergodic measure preserving action $Gamma curvearrowright (X,mu)$ of a discrete group $Gamma$ on a $sigma$-finite measure space $(X,mu)$ satisfies the local spectral gap property (introduced by Boutonnet, Ioana and Salehi Golsefidy) if and only if it is strongly ergodic. In fact, we prove a more general local spectral gap criterion in arbitrary von Neumann algebras. Using this criterion, we also obtain a short and elementary proof of Connes' spectral gap theorem for full $mathrm{II}_1$ factors as well as its recent generalization to full type $mathrm{III}$ factors.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"186 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124717675","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}
In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning actions of locally compact quantum groups on C*-algebras [S. Baaj, G. Skandalis and S. Vaes, 2003]. Let $cal G$ be a measured quantum groupoid on a finite basis. We prove that if $cal G$ is regular, then any weakly continuous action of $cal G$ on a C*-algebra is necessarily strongly continuous. Following [S. Baaj and G. Skandalis, 1989], we introduce and investigate a notion of $cal G$-equivariant Hilbert C$^*$-modules. By applying the previous results and a version of the Takesaki-Takai duality theorem obtained in [S. Baaj and J. C., 2015] for actions of $cal G$, we obtain a canonical equivariant Morita equivalence between a given $cal G$-C$^*$-algebra $A$ and the double crossed product $(Artimes{cal G})rtimeswidehat{cal G}$.
本文将局部紧量子群在C*-代数上作用的一些重要结果推广到有限基上实测量子群的情况。[j].科学与技术，2003。设$ G$为有限基上的可测量子群。证明了如果$cal G$是正则的，则$cal G$在C*-代数上的任何弱连续作用必然是强连续的。[S。Baaj and G. Skandalis, 1989]，我们引入并研究了$cal $-等变Hilbert C$^*$-模的概念。应用前人的结果和[S]中得到的Takesaki-Takai对偶定理的一个版本。Baaj and J. C.， 2015]对于$cal G$的作用，我们得到了给定$cal G$-C$^*$-代数$ a $与双交叉积$(a rtimes{cal G})rtimeswidehat{cal G}$之间的正则等变Morita等价。
{"title":"Actions of measured quantum groupoids on a finite basis","authors":"Jonathan Crespo","doi":"10.1215/ijm/1552442659","DOIUrl":"https://doi.org/10.1215/ijm/1552442659","url":null,"abstract":"In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning actions of locally compact quantum groups on C*-algebras [S. Baaj, G. Skandalis and S. Vaes, 2003]. Let $cal G$ be a measured quantum groupoid on a finite basis. We prove that if $cal G$ is regular, then any weakly continuous action of $cal G$ on a C*-algebra is necessarily strongly continuous. Following [S. Baaj and G. Skandalis, 1989], we introduce and investigate a notion of $cal G$-equivariant Hilbert C$^*$-modules. By applying the previous results and a version of the Takesaki-Takai duality theorem obtained in [S. Baaj and J. C., 2015] for actions of $cal G$, we obtain a canonical equivariant Morita equivalence between a given $cal G$-C$^*$-algebra $A$ and the double crossed product $(Artimes{cal G})rtimeswidehat{cal G}$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"277 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133383004","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 use non-symmetric distances to give a self-contained account of C*-algebra filters and their corresponding compact projections, simultaneously simplifying and extending their general theory.
我们利用非对称距离给出了C*-代数滤波器及其紧投影的自包含说明，同时简化和扩展了它们的一般理论。
{"title":"C*-Algebra Distance Filters","authors":"T. Bice, A. Vignati","doi":"10.15352/aot.1710-1241","DOIUrl":"https://doi.org/10.15352/aot.1710-1241","url":null,"abstract":"We use non-symmetric distances to give a self-contained account of C*-algebra filters and their corresponding compact projections, simultaneously simplifying and extending their general theory.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"49 52","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120836906","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 arbitrary operator ideal I, every nilpotent element of I is a single commutator of operators from I^t, for an exponent t that depends on the degree of nilpotency.
对于任意算子理想I, I的每一个幂零元素都是I^t中算子的对易子，对于指数t，它取决于幂零的程度。
{"title":"Nilpotent elements of operator ideals as single commutators","authors":"K. Dykema, Amudhan Krishnaswamy-Usha","doi":"10.1090/proc/13987","DOIUrl":"https://doi.org/10.1090/proc/13987","url":null,"abstract":"For an arbitrary operator ideal I, every nilpotent element of I is a single commutator of operators from I^t, for an exponent t that depends on the degree of nilpotency.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"202 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121090977","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}
Pub Date : 2017-06-02DOI: 10.7900/JOT.2017SEP28.2192
A. Kumjian, D. Pask, A. Sims
We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner-Voiculescu sequence. We introduce the notion of a twisted P-graph C*-algebra and establish connections with graded C*-algebras. Specifically, we show how a functor from a P-graph into the group of order two determines a grading of the associated C*-algebra. We apply our graded version of Pimsner's exact sequence to compute the graded K-theory of a graph C*-algebra carrying such a grading.
{"title":"Graded C*-algebras, Graded K-theory, And Twisted P-graph C*-algebras","authors":"A. Kumjian, D. Pask, A. Sims","doi":"10.7900/JOT.2017SEP28.2192","DOIUrl":"https://doi.org/10.7900/JOT.2017SEP28.2192","url":null,"abstract":"We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner-Voiculescu sequence. We introduce the notion of a twisted P-graph C*-algebra and establish connections with graded C*-algebras. Specifically, we show how a functor from a P-graph into the group of order two determines a grading of the associated C*-algebra. We apply our graded version of Pimsner's exact sequence to compute the graded K-theory of a graph C*-algebra carrying such a grading.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130418264","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}
Pub Date : 2017-05-19DOI: 10.1142/S0219025718500054
A. Talebi, M. Moslehian, G. Sadeghi
We establish a noncommutative Blackwell--Ross inequality for supermartingales under a suitable condition which generalize Khan's works to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.
{"title":"Noncommutative Blackwell-Ross martingale inequality","authors":"A. Talebi, M. Moslehian, G. Sadeghi","doi":"10.1142/S0219025718500054","DOIUrl":"https://doi.org/10.1142/S0219025718500054","url":null,"abstract":"We establish a noncommutative Blackwell--Ross inequality for supermartingales under a suitable condition which generalize Khan's works to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134334323","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}
Pub Date : 2017-05-13DOI: 10.1215/00127094-2020-0034
G. Hong, Benben Liao, Simeng Wang
This paper is devoted to the study of noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $alpha$ be a continuous action of $G$ on a von Neumann algebra $mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by [ A_{n}x=frac{1}{m(V^{n})}int_{V^{n}}alpha_{g}xdm(g),quad xin L_{p}(mathcal{M}),ninmathbb{N},1leq pleq infty ] is of weak type $(1,1)$ and of strong type $(p,p)$ for $1 < p
本文研究了von Neumann代数上群作用的非交换极大不等式和遍历定理。考虑一个多项式增长的局部紧群$G$和一个对称紧子集$V$。设$alpha$为保持迹自同构在冯·诺依曼代数$mathcal{M}$上的连续作用$G$。然后我们证明[ A_{n}x=frac{1}{m(V^{n})}int_{V^{n}}alpha_{g}xdm(g),quad xin L_{p}(mathcal{M}),ninmathbb{N},1leq pleq infty ]定义的运算符对于$1 < p
{"title":"Noncommutative maximal ergodic inequalities associated with doubling conditions","authors":"G. Hong, Benben Liao, Simeng Wang","doi":"10.1215/00127094-2020-0034","DOIUrl":"https://doi.org/10.1215/00127094-2020-0034","url":null,"abstract":"This paper is devoted to the study of noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact group $G$ of polynomial growth with a symmetric compact subset $V$. Let $alpha$ be a continuous action of $G$ on a von Neumann algebra $mathcal{M}$ by trace-preserving automorphisms. We then show that the operators defined by [ A_{n}x=frac{1}{m(V^{n})}int_{V^{n}}alpha_{g}xdm(g),quad xin L_{p}(mathcal{M}),ninmathbb{N},1leq pleq infty ] is of weak type $(1,1)$ and of strong type $(p,p)$ for $1 < p<infty$. Consequently, the sequence $(A_{n}x)_{ngeq 1}$ converges almost uniformly for $xin L_{p}(mathcal{M})$ for $1leq p<infty$. Also we establish the noncommutative maximal and individual ergodic theorems associated with more general doubling conditions; and we prove the corresponding results for general actions on one fixed noncommutative $L_p$-space which are beyond the class of Dunford-Schwartz operators considered previously by Junge and Xu. As key ingredients, we also obtain the Hardy-Littlewood maximal inequality on metric spaces with doubling measures in the operator-valued setting. After the groundbreaking work of Junge and Xu on the noncommutative Dunford-Schwartz maximal ergodic inequalities, this is the first time that more general maximal inequalities are proved beyond Junge-Xu's setting. Our approach is based on the quantum probabilistic methods as well as the random walk theory.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115702424","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 work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $Cleft( mathbb{P}^{n}left( mathcal{T}right) right) $ and $Cleft( mathbb{S}_{H}^{2n+1}right) $ of the quantum complex projective spaces $mathbb{P}^{n}left( mathcal{T} right) $ and the quantum spheres $mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $mathbb{P}^{n}left( mathcal{T}right) $, studied by Hajac and collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault to the study of C*-algebraic structure, we analyze $Cleft( mathbb{P}^{n}left( mathcal{T}right) right) $, $Cleft( mathbb{S}_{H}^{2n+1}right) $, and $L_{k}$ in the context of groupoid C*-algebras, and then apply Rieffel's stable rank results to show that all finitely generated projective modules over $Cleft( mathbb{S}_{H} ^{2n+1}right) $ of rank higher than $leftlfloor frac{n}{2}rightrfloor +3$ are free modules. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group of $Cleft( mathbb{P}^{n}left( mathcal{T}right) right) $, we also explicitly identify $L_{k}$ with concrete representative elementary projections over $Cleft( mathbb{P} ^{n}left( mathcal{T}right) right) $.
{"title":"Vector bundles over multipullback quantum complex projective spaces","authors":"A. Sheu","doi":"10.4171/jncg/401","DOIUrl":"https://doi.org/10.4171/jncg/401","url":null,"abstract":"We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $Cleft( mathbb{P}^{n}left( mathcal{T}right) right) $ and $Cleft( mathbb{S}_{H}^{2n+1}right) $ of the quantum complex projective spaces $mathbb{P}^{n}left( mathcal{T} right) $ and the quantum spheres $mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $mathbb{P}^{n}left( mathcal{T}right) $, studied by Hajac and collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault to the study of C*-algebraic structure, we analyze $Cleft( mathbb{P}^{n}left( mathcal{T}right) right) $, $Cleft( mathbb{S}_{H}^{2n+1}right) $, and $L_{k}$ in the context of groupoid C*-algebras, and then apply Rieffel's stable rank results to show that all finitely generated projective modules over $Cleft( mathbb{S}_{H} ^{2n+1}right) $ of rank higher than $leftlfloor frac{n}{2}rightrfloor +3$ are free modules. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group of $Cleft( mathbb{P}^{n}left( mathcal{T}right) right) $, we also explicitly identify $L_{k}$ with concrete representative elementary projections over $Cleft( mathbb{P} ^{n}left( mathcal{T}right) right) $.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114094109","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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