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Nuclear dimension and virtually polycyclic groups 核维度和实际多环群
arXiv - MATH - Operator Algebras Pub Date : 2024-08-13 DOI: arxiv-2408.07223
Caleb Eckhardt, Jianchao Wu
{"title":"Nuclear dimension and virtually polycyclic groups","authors":"Caleb Eckhardt, Jianchao Wu","doi":"arxiv-2408.07223","DOIUrl":"https://doi.org/arxiv-2408.07223","url":null,"abstract":"We show that the nuclear dimension of a (twisted) group C*-algebra of a\u0000virtually polycyclic group is finite. This prompts us to make a conjecture\u0000relating finite nuclear dimension of group C*-algebras and finite Hirsch\u0000length, which we then verify for a class of elementary amenable groups beyond\u0000the virtually polycyclic case. In particular, we give the first examples of\u0000finitely generated, non-residually finite groups with finite nuclear dimension.\u0000A parallel conjecture on finite decomposition rank is also formulated and an\u0000analogous result is obtained. Our method relies heavily on recent work of\u0000Hirshberg and the second named author on actions of virtually nilpotent groups\u0000on $C_0(X)$-algebras.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195101","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
The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure 允许 A-by-FCE 粗纤维结构的空间的最大粗鲍姆-康内斯猜想
arXiv - MATH - Operator Algebras Pub Date : 2024-08-13 DOI: arxiv-2408.06660
Liang Guo, Qin Wang, Chen Zhang
{"title":"The maximal coarse Baum-Connes conjecture for spaces that admit an A-by-FCE coarse fibration structure","authors":"Liang Guo, Qin Wang, Chen Zhang","doi":"arxiv-2408.06660","DOIUrl":"https://doi.org/arxiv-2408.06660","url":null,"abstract":"In this paper, we introduce the concept of an A-by-FCE coarse fibration\u0000structure for metric spaces, which serves as a generalization of the A-by-CE\u0000structure for a sequence of group extensions proposed by Deng, Wang, and Yu. We\u0000show that the maximal coarse Baum-Connes conjecture holds for metric spaces\u0000with bounded geometry that admit an A-by-FCE coarse fibration structure. As an\u0000application, the relative expanders constructed by Arzhantseva and Tessera, as\u0000well as the box space derived from an extension of Haagerup groups by amenable\u0000groups, are shown to exhibit the A-by-FCE coarse fibration structure.\u0000Consequently, their maximal coarse Baum-Connes conjectures are affirmed.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195067","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
Operator means, barycenters, and fixed point equations 运算符手段、原点和定点方程
arXiv - MATH - Operator Algebras Pub Date : 2024-08-12 DOI: arxiv-2408.06343
Dániel Virosztek
{"title":"Operator means, barycenters, and fixed point equations","authors":"Dániel Virosztek","doi":"arxiv-2408.06343","DOIUrl":"https://doi.org/arxiv-2408.06343","url":null,"abstract":"The seminal work of Kubo and Ando from 1980 provided us with an axiomatic\u0000approach to means of positive operators. As most of their axioms are algebraic\u0000in nature, this approach has a clear algebraic flavor. On the other hand, it is\u0000highly natural to take the geometric viewpoint and consider a distance\u0000(understood in a broad sense) on the cone of positive operators, and define the\u0000mean of positive operators by an appropriate notion of the center of mass. This\u0000strategy often leads to a fixed point equation that characterizes the mean. The\u0000aim of this survey is to highlight those cases where the algebraic and the\u0000geometric approaches meet each other.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195071","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
Rigid Graph Products 硬质图形产品
arXiv - MATH - Operator Algebras Pub Date : 2024-08-12 DOI: arxiv-2408.06171
Matthijs Borst, Martijn Caspers, Enli Chen
{"title":"Rigid Graph Products","authors":"Matthijs Borst, Martijn Caspers, Enli Chen","doi":"arxiv-2408.06171","DOIUrl":"https://doi.org/arxiv-2408.06171","url":null,"abstract":"We prove rigidity properties for von Neumann algebraic graph products. We\u0000introduce the notion of rigid graphs and define a class of II$_1$-factors named\u0000$mathcal{C}_{rm Rigid}$. For von Neumann algebras in this class we show a\u0000unique rigid graph product decomposition. In particular, we obtain unique prime\u0000factorization results and unique free product decomposition results for new\u0000classes of von Neumann algebras. We also prove several technical results\u0000concerning relative amenability and embeddings of (quasi)-normalizers in graph\u0000products. Furthermore, we give sufficient conditions for a graph product to be\u0000nuclear and characterize strong solidity, primeness and free-indecomposability\u0000for graph products.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195068","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
On the Completely Positive Approximation Property for Non-Unital Operator Systems and the Boundary Condition for the Zero Map 论非无算子系统的完全正逼近性质和零图的边界条件
arXiv - MATH - Operator Algebras Pub Date : 2024-08-12 DOI: arxiv-2408.06127
Se-Jin Kim
{"title":"On the Completely Positive Approximation Property for Non-Unital Operator Systems and the Boundary Condition for the Zero Map","authors":"Se-Jin Kim","doi":"arxiv-2408.06127","DOIUrl":"https://doi.org/arxiv-2408.06127","url":null,"abstract":"The purpose of this paper is two-fold: firstly, we give a characterization on\u0000the level of non-unital operator systems for when the zero map is a boundary\u0000representation. As a consequence, we show that a non-unital operator system\u0000arising from the direct limit of C*-algebras under positive maps is a\u0000C*-algebra if and only if its unitization is a C*-algebra. Secondly, we show\u0000that the completely positive approximation property and the completely\u0000contractive approximation property of a non-unital operator system is\u0000equivalent to its bidual being an injective von Neumann algebra. This implies\u0000in particular that all non-unital operator systems with the completely\u0000contractive approximation property must necessarily admit an abundance of\u0000positive elements.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195069","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
A noncommutative maximal inequality for ergodic averages along arithmetic sets 沿算术集合的遍历平均的非交换最大不等式
arXiv - MATH - Operator Algebras Pub Date : 2024-08-08 DOI: arxiv-2408.04374
Cheng Chen, Guixiang Hong, Liang Wang
{"title":"A noncommutative maximal inequality for ergodic averages along arithmetic sets","authors":"Cheng Chen, Guixiang Hong, Liang Wang","doi":"arxiv-2408.04374","DOIUrl":"https://doi.org/arxiv-2408.04374","url":null,"abstract":"In this paper, we establish a noncommutative maximal inequality for ergodic\u0000averages with respect to the set ${k^t|k=1,2,3,...}$ acting on noncommutative\u0000$L_p$ spaces for $p>frac{sqrt{5}+1}{2}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930501","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
Quantum metric Choquet simplices 量子度量乔凯简约
arXiv - MATH - Operator Algebras Pub Date : 2024-08-08 DOI: arxiv-2408.04368
Bhishan Jacelon
{"title":"Quantum metric Choquet simplices","authors":"Bhishan Jacelon","doi":"arxiv-2408.04368","DOIUrl":"https://doi.org/arxiv-2408.04368","url":null,"abstract":"Precipitating a notion emerging from recent research, we formalise the study\u0000of a special class of compact quantum metric spaces. Abstractly, the additional\u0000requirement we impose on the underlying order unit spaces is the Riesz\u0000interpolation property. In practice, this means that a `quantum metric Choquet\u0000simplex' arises as a unital $mathrm{C}^*$-algebra $A$ whose trace space is\u0000equipped with a metric inducing the $w^*$-topology, such that tracially\u0000Lipschitz elements are dense in $A$. This added structure is designed for\u0000measuring distances in and around the category of stably finite classifiable\u0000$mathrm{C}^*$-algebras, and in particular for witnessing metric and\u0000statistical properties of the space of (approximate unitary equivalence classes\u0000of) unital embeddings of $A$ into a stably finite classifiable\u0000$mathrm{C}^*$-algebra $B$. Our reference frame for this measurement is a\u0000certain compact `nucleus' of $A$ provided by its quantum metric structure. As\u0000for the richness of the metric space of isometric isomorphism classes of\u0000classifiable $mathrm{C}^*$-algebraic quantum metric Choquet simplices\u0000(equipped with Rieffel's quantum Gromov--Hausdorff distance), we show how to\u0000construct examples starting from Bauer simplices associated to compact metric\u0000spaces. We also explain how to build non-Bauer examples by forming `quantum\u0000crossed products' associated to dynamical systems on the tracial boundary.\u0000Further, we observe that continuous fields of quantum spaces are obtained by\u0000continuously varying either the dynamics or the metric. In the case of deformed\u0000isometric actions, we show that equivariant Gromov--Hausdorff continuity\u0000implies fibrewise continuity of the quantum structures. As an example, we\u0000present a field of deformed quantum rotation algebras whose fibres are\u0000continuous with respect to a quasimetric called the quantum intertwining gap.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930552","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
Criteria for the existence of Schwartz Gabor frames over rational lattices 有理网格上施瓦茨 Gabor 框架的存在标准
arXiv - MATH - Operator Algebras Pub Date : 2024-08-06 DOI: arxiv-2408.03423
Ulrik Enstad, Hannes Thiel, Eduard Vilalta
{"title":"Criteria for the existence of Schwartz Gabor frames over rational lattices","authors":"Ulrik Enstad, Hannes Thiel, Eduard Vilalta","doi":"arxiv-2408.03423","DOIUrl":"https://doi.org/arxiv-2408.03423","url":null,"abstract":"We give an explicit criterion for a rational lattice in the time-frequency\u0000plane to admit a Gabor frame with window in the Schwartz class. The criterion\u0000is an inequality formulated in terms of the lattice covolume, the dimension of\u0000the underlying Euclidean space, and the index of an associated subgroup\u0000measuring the degree of non-integrality of the lattice. For arbitrary lattices\u0000we also give an upper bound on the number of windows in the Schwartz class\u0000needed for a multi-window Gabor frame.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930509","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
Large Perturbations of Nest Algebras 巢代数的大扰动
arXiv - MATH - Operator Algebras Pub Date : 2024-08-06 DOI: arxiv-2408.03317
Kenneth R. Davidson
{"title":"Large Perturbations of Nest Algebras","authors":"Kenneth R. Davidson","doi":"arxiv-2408.03317","DOIUrl":"https://doi.org/arxiv-2408.03317","url":null,"abstract":"Let $mathcal{M}$ and $mathcal{N}$ be nests on separable Hilbert space. If\u0000the two nest algebras are distance less than 1\u0000($d(mathcal{T}(mathcal{M}),mathcal{T}(mathcal{N})) < 1$), then the nests\u0000are distance less than 1 ($d(mathcal{M},mathcal{N})<1$). If the nests are\u0000distance less than 1 apart, then the nest algebras are similar, i.e. there is\u0000an invertible $S$ such that $Smathcal{M} = mathcal{N}$, so that $S\u0000mathcal{T}(mathcal{M})S^{-1} = mathcal{T}(mathcal{N})$. However there are\u0000examples of nests closer than 1 for which the nest algebras are distance 1\u0000apart.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930508","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
Groups acting amenably on their Higson corona 和睦相处的希格森日冕团体
arXiv - MATH - Operator Algebras Pub Date : 2024-08-06 DOI: arxiv-2408.02997
Alexander Engel
{"title":"Groups acting amenably on their Higson corona","authors":"Alexander Engel","doi":"arxiv-2408.02997","DOIUrl":"https://doi.org/arxiv-2408.02997","url":null,"abstract":"We investigate groups that act amenably on their Higson corona (also known as\u0000bi-exact groups) and we provide reformulations of this in relation to the\u0000stable Higson corona, nuclearity of crossed products and to positive type\u0000kernels. We further investigate implications of this in relation to the\u0000Baum-Connes conjecture, and prove that Gromov hyperbolic groups have isomorphic\u0000equivariant K-theories of their Gromov boundary and their stable Higson corona.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930589","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
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