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arXiv: Operator Algebras最新文献

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The covariant Gromov–Hausdorff propinquity 协变Gromov-Hausdorff近似
arXiv: Operator Algebras Pub Date : 2018-05-29 DOI: 10.4064/sm180610-28-12
F. Latrémolière
{"title":"The covariant Gromov–Hausdorff propinquity","authors":"F. Latrémolière","doi":"10.4064/sm180610-28-12","DOIUrl":"https://doi.org/10.4064/sm180610-28-12","url":null,"abstract":"We extend the Gromov-Hausdorff propinquity to a metric on Lipschitz dynamical systems, which are given by strongly continuous actions of proper monoids on quantum compact metric spaces via Lipschitz morphisms. We prove that our resulting metric is zero between two Lipschitz dynamical systems if and only if there exists an equivariant full quantum isometry between. We also present sufficient conditions for Cauchy sequences to converge for our new metric, thus exhibiting certain complete classes of Lipschitz dynamical systems. We apply our work to convergence of the dual actions on fuzzy tori to the dual actions on quantum tori. Our framework is general enough to also allow for the study of the convergence of continuous semigroups of positive linear maps and other actions of proper monoids.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"601 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133432464","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: 15
An elementary approach to free entropy theory for convex potentials 凸势自由熵理论的基本方法
arXiv: Operator Algebras Pub Date : 2018-05-22 DOI: 10.2140/apde.2020.13.2289
David Jekel
{"title":"An elementary approach to free entropy theory for convex potentials","authors":"David Jekel","doi":"10.2140/apde.2020.13.2289","DOIUrl":"https://doi.org/10.2140/apde.2020.13.2289","url":null,"abstract":"We present an alternative approach to the theory of free Gibbs states with convex potentials developed in several papers of Guionnet, Shlyakhtenko, and Dabrowski. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(mathbb{C})_{sa}^m$ to prove the following. Suppose $mu_N$ is a probability measure on on $M_N(mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials. Then the moments of $mu_N$ converge to a non-commutative law $lambda$. Moreover, the free entropies $chi(lambda)$, $underline{chi}(lambda)$, and $chi^*(lambda)$ agree and equal the limit of the normalized classical entropies of $mu_N$. A key step is to show that the property of asymptotic approximation by trace polynomials is preserved under several operations, including limits, composition, Gaussian convolution, and ultimately evolution under certain parabolic PDE. This allows us to prove convergence of the moments of $mu_N$ and of the Fisher information of Gaussian perturbations of $mu_N$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132761977","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: 10
On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces 一致局部有限粗糙空间上一致Roe代数的刚性
arXiv: Operator Algebras Pub Date : 2018-05-11 DOI: 10.1090/tran/8180
B. M. Braga, I. Farah
{"title":"On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces","authors":"B. M. Braga, I. Farah","doi":"10.1090/tran/8180","DOIUrl":"https://doi.org/10.1090/tran/8180","url":null,"abstract":"Given a coarse space $(X,mathcal{E})$, one can define a $mathrm{C}^*$-algebra $mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,mathcal{E})$. It has been proved by J. v{S}pakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126218128","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: 21
Compact group actions on operator algebras and their spectra 算子代数上的紧群作用及其谱
arXiv: Operator Algebras Pub Date : 2018-04-24 DOI: 10.7146/math.scand.a-119079
C. Peligrad
{"title":"Compact group actions on operator algebras and their spectra","authors":"C. Peligrad","doi":"10.7146/math.scand.a-119079","DOIUrl":"https://doi.org/10.7146/math.scand.a-119079","url":null,"abstract":"We consider a class of dynamical systems with compact non abelian groups that include C*-, W*- and multiplier dynamical systems. We prove results that relate the algebraic properties such as simplicity or primeness of the fixed point algebras as defned in Section 3., to the spectral properties of the action, including the Connes and strong Connes spectra.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127770178","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
Spectral triples for higher-rank graph $C^*$-algebras 高阶图C^* -代数的谱三元组
arXiv: Operator Algebras Pub Date : 2018-04-14 DOI: 10.7146/math.scand.a-119260
Carla Farsi, E. Gillaspy, A. Julien, Sooran Kang, J. Packer
{"title":"Spectral triples for higher-rank graph $C^*$-algebras","authors":"Carla Farsi, E. Gillaspy, A. Julien, Sooran Kang, J. Packer","doi":"10.7146/math.scand.a-119260","DOIUrl":"https://doi.org/10.7146/math.scand.a-119260","url":null,"abstract":"In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(Lambda)$ of a strongly connected finite higher-rank graph $Lambda$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(Lambda)$, and we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $Lambda$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of this spectral triple.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122253139","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
Characterizing projections among positive operators in the unit sphere 单位球面上正算子间投影的刻画
arXiv: Operator Algebras Pub Date : 2018-04-11 DOI: 10.15352/AOT.1804-1343
A. M. Peralta
{"title":"Characterizing projections among positive operators in the unit sphere","authors":"A. M. Peralta","doi":"10.15352/AOT.1804-1343","DOIUrl":"https://doi.org/10.15352/AOT.1804-1343","url":null,"abstract":"Let $E$ and $P$ be subsets of a Banach space $X$, and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P) :=left{ xin P : |x-b|=1 hbox{ for all } bin E right}.$$ Given a C$^*$-algebra $A$, and a subset $Esubset A,$ we shall write $Sph^+ (E)$ or $Sph_A^+ (E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ stands for the set of all positive operators in the unit sphere of $A$. We prove that, for an arbitrary complex Hilbert space $H$, then a positive element $a$ in the unit sphere of $B(H)$ is a projection if and only if $Sph^+_{B(H)} left( Sph^+_{B(H)}({a}) right) ={a}$. We also prove that the equivalence remains true when $B(H)$ is replaced with an atomic von Neumann algebra or with $K(H_2)$, where $H_2$ is an infinite-dimensional and separable complex Hilbert space. In the setting of compact operators we prove a stronger conclusion by showing that the identity $$Sph^+_{K(H_2)} left( Sph^+_{K(H_2)}(a) right) =left{ bin S(K(H_2)^+) : !! begin{array}{c} s_{_{K(H_2)}} (a) leq s_{_{K(H_2)}} (b), hbox{ and } textbf{1}-r_{_{B(H_2)}}(a)leq textbf{1}-r_{_{B(H_2)}}(b) end{array}!! right},$$ holds for every $a$ in the unit sphere of $K(H_2)^+$, where $r_{_{B(H_2)}}(a)$ and $s_{_{K(H_2)}} (a)$ stand for the range and support projections of $a$ in $B(H_2)$ and $K(H_2)$, respectively.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133758640","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: 4
Three characterisations of the sequential product 序列积的三个特征
arXiv: Operator Algebras Pub Date : 2018-03-13 DOI: 10.1063/1.5031089
J. V. D. Wetering
{"title":"Three characterisations of the sequential product","authors":"J. V. D. Wetering","doi":"10.1063/1.5031089","DOIUrl":"https://doi.org/10.1063/1.5031089","url":null,"abstract":"It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product $acirc b = sqrt{a}bsqrt{a}$ on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations we first have to study convex $sigma$-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126554284","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: 11
Invertibility of Generalized Bessel multipliers in Hilbert $C^{*}$-modules Hilbert $C^{*}$-模中广义贝塞尔乘子的可逆性
arXiv: Operator Algebras Pub Date : 2018-02-06 DOI: 10.4134/BKMS.B200358
G. Tabadkan, Hessam Hossein-nezhad
{"title":"Invertibility of Generalized Bessel multipliers in Hilbert $C^{*}$-modules","authors":"G. Tabadkan, Hessam Hossein-nezhad","doi":"10.4134/BKMS.B200358","DOIUrl":"https://doi.org/10.4134/BKMS.B200358","url":null,"abstract":"In this note, a general version of Bessel multipliers in Hilbert $C^*$-modules is presented and then, many results obtained for multipliers are extended. Also the conditions for invertibility of generalized multipliers are investigated in details. The invertibility of multipliers is very important because it helps us to obtain more reconstruction formula.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"156 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122877761","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
Simple equivariant C*-algebras whose full and reduced crossed products coincide 其满积和约简积重合的简单等变C*代数
arXiv: Operator Algebras Pub Date : 2018-01-22 DOI: 10.4171/JNCG/356
Yuhei Suzuki
{"title":"Simple equivariant C*-algebras whose full and reduced crossed products coincide","authors":"Yuhei Suzuki","doi":"10.4171/JNCG/356","DOIUrl":"https://doi.org/10.4171/JNCG/356","url":null,"abstract":"For any second countable locally compact group G, we construct a simple G-C*-algebra whose full and reduced crossed product norms coincide. We then construct its G-equivariant representation on another simple G-C*-algebra without the coincidence condition. This settles two problems posed by Anantharaman-Delaroche in 2002. Some constructions involve the Baire category theorem.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122449682","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: 21
An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras 有限von Neumann代数下非交换Hardy空间的Beurling-Chen-Hadwin-Shen定理的推广
arXiv: Operator Algebras Pub Date : 2018-01-14 DOI: 10.7153/OAM-2020-14-49
Haihui Fan, D. Hadwin, Wenjing Liu
{"title":"An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras","authors":"Haihui Fan, D. Hadwin, Wenjing Liu","doi":"10.7153/OAM-2020-14-49","DOIUrl":"https://doi.org/10.7153/OAM-2020-14-49","url":null,"abstract":"In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling's theorems for a continuous unitarily invariant norm $% alpha $ on a tracial von Neumann algebra $left( mathcal{M},tau right) $ where $alpha $ is $leftVert cdot rightVert _{1}$-dominating with respect to $tau $. In the paper, we first define a class of norms $% N_{Delta }left( mathcal{M},tau right) $ on $mathcal{M}$, called determinant, normalized, unitarily invariant continuous norms on $mathcal{M}$. If $alpha in N_{Delta }left( mathcal{M},tau right) $, then there exists a faithful normal tracial state $rho $ on $mathcal{M}$ such that $rho left( xright) =tau left( xgright) $ for some positive $gin L^{1}left( mathcal{Z},tau right) $ and the determinant of $g$ is positive. For every $alpha in N_{Delta }left( mathcal{M},tau right) $, we study the noncommutative Hardy spaces $% H^{alpha }left( mathcal{M},tau right) $, then prove that the Chen-Hadwin-Shen theorem holds for $L^{alpha }left( mathcal{M},tau right) $. The key ingredients in the proof of our result include a factorization theorem and a density theorem for $L^{alpha }left( mathcal{M},rho right) $.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127279793","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: 1
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