We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on $C_0(X)$-algebras.
{"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":"75 1","pages":""},"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}
In this paper, we introduce the concept of an A-by-FCE coarse fibration structure for metric spaces, which serves as a generalization of the A-by-CE structure for a sequence of group extensions proposed by Deng, Wang, and Yu. We show that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry that admit an A-by-FCE coarse fibration structure. As an application, the relative expanders constructed by Arzhantseva and Tessera, as well as the box space derived from an extension of Haagerup groups by amenable groups, are shown to exhibit the A-by-FCE coarse fibration structure. Consequently, their maximal coarse Baum-Connes conjectures are affirmed.
{"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":"108 1","pages":""},"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}
The seminal work of Kubo and Ando from 1980 provided us with an axiomatic approach to means of positive operators. As most of their axioms are algebraic in nature, this approach has a clear algebraic flavor. On the other hand, it is highly natural to take the geometric viewpoint and consider a distance (understood in a broad sense) on the cone of positive operators, and define the mean of positive operators by an appropriate notion of the center of mass. This strategy often leads to a fixed point equation that characterizes the mean. The aim of this survey is to highlight those cases where the algebraic and the geometric approaches meet each other.
{"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":"108 1","pages":""},"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}
We prove rigidity properties for von Neumann algebraic graph products. We introduce the notion of rigid graphs and define a class of II$_1$-factors named $mathcal{C}_{rm Rigid}$. For von Neumann algebras in this class we show a unique rigid graph product decomposition. In particular, we obtain unique prime factorization results and unique free product decomposition results for new classes of von Neumann algebras. We also prove several technical results concerning relative amenability and embeddings of (quasi)-normalizers in graph products. Furthermore, we give sufficient conditions for a graph product to be nuclear and characterize strong solidity, primeness and free-indecomposability for graph products.
我们证明了 von Neumann 代数图积的刚性属性。我们引入了刚性图的概念，并定义了一类名为$mathcal{C}_{rm Rigid}$的 II$_1$ 因子。对于这一类中的冯-诺依曼代数，我们展示了独特的刚性图积分解。特别是，我们得到了新类冯-诺依曼代数的唯一素因子化结果和唯一自由积分解结果。我们还证明了与图积中的（准）归一化子的相对适配性和嵌入有关的几个技术结果。此外，我们还给出了图积成核的充分条件，并描述了图积的强实体性、原始性和自由不可分性。
{"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":"26 1","pages":""},"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}
The purpose of this paper is two-fold: firstly, we give a characterization on the level of non-unital operator systems for when the zero map is a boundary representation. As a consequence, we show that a non-unital operator system arising from the direct limit of C*-algebras under positive maps is a C*-algebra if and only if its unitization is a C*-algebra. Secondly, we show that the completely positive approximation property and the completely contractive approximation property of a non-unital operator system is equivalent to its bidual being an injective von Neumann algebra. This implies in particular that all non-unital operator systems with the completely contractive approximation property must necessarily admit an abundance of positive elements.
{"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":"14 1","pages":""},"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}
In this paper, we establish a noncommutative maximal inequality for ergodic averages with respect to the set ${k^t|k=1,2,3,...}$ acting on noncommutative $L_p$ spaces for $p>frac{sqrt{5}+1}{2}$.
{"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":"30 1","pages":""},"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}
Precipitating a notion emerging from recent research, we formalise the study of a special class of compact quantum metric spaces. Abstractly, the additional requirement we impose on the underlying order unit spaces is the Riesz interpolation property. In practice, this means that a `quantum metric Choquet simplex' arises as a unital $mathrm{C}^*$-algebra $A$ whose trace space is equipped with a metric inducing the $w^*$-topology, such that tracially Lipschitz elements are dense in $A$. This added structure is designed for measuring distances in and around the category of stably finite classifiable $mathrm{C}^*$-algebras, and in particular for witnessing metric and statistical properties of the space of (approximate unitary equivalence classes of) unital embeddings of $A$ into a stably finite classifiable $mathrm{C}^*$-algebra $B$. Our reference frame for this measurement is a certain compact `nucleus' of $A$ provided by its quantum metric structure. As for the richness of the metric space of isometric isomorphism classes of classifiable $mathrm{C}^*$-algebraic quantum metric Choquet simplices (equipped with Rieffel's quantum Gromov--Hausdorff distance), we show how to construct examples starting from Bauer simplices associated to compact metric spaces. We also explain how to build non-Bauer examples by forming `quantum crossed products' associated to dynamical systems on the tracial boundary. Further, we observe that continuous fields of quantum spaces are obtained by continuously varying either the dynamics or the metric. In the case of deformed isometric actions, we show that equivariant Gromov--Hausdorff continuity implies fibrewise continuity of the quantum structures. As an example, we present a field of deformed quantum rotation algebras whose fibres are continuous with respect to a quasimetric called the quantum intertwining gap.
根据最近研究中出现的一个概念，我们将对一类特殊的紧凑量子度量空间的研究形式化。抽象地说，我们对底层阶单元空间施加的额外要求是里兹插值特性。在实践中，这意味着 "量子度量乔奎兹复数 "是作为一个单价$mathrm{C}^*$-代数$A$而产生的，其痕量空间配备了一个诱导$w^*$拓扑的度量，使得痕量利普希兹元素在$A$中是密集的。这个新增结构旨在测量稳定有限可分类$mathrm{C}^*$代数范畴内及其周围的距离，特别是用于见证将$A$嵌入稳定有限可分类$mathrm{C}^*$代数$B$的（近似单元等价类的）单元嵌入空间的度量和统计性质。我们测量的参照系是由量子度量结构提供的$A$的某个紧凑 "核"。为了丰富可分类 $mathrm{C}^*$ 代数量子度量乔凯简约（配备里费尔量子格罗莫夫--豪斯多夫距离）的等距同构类的度量空间，我们展示了如何从与紧凑度量空间相关的鲍尔简约开始构建例子。此外，我们还解释了如何通过形成与三维边界上的动力系统相关的 "量子交叉积 "来建立非鲍尔范例。在变形等距作用的情况下，我们证明等变格罗莫夫-豪斯多夫连续性意味着量子结构的纤维连续性。举例来说，我们提出了一个变形量子旋转代数场，它的纤维相对于量子交织间隙（quasimetric called the quantum intertwining gap）是连续的。
{"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":"56 1","pages":""},"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}
We give an explicit criterion for a rational lattice in the time-frequency plane to admit a Gabor frame with window in the Schwartz class. The criterion is an inequality formulated in terms of the lattice covolume, the dimension of the underlying Euclidean space, and the index of an associated subgroup measuring the degree of non-integrality of the lattice. For arbitrary lattices we also give an upper bound on the number of windows in the Schwartz class needed for a multi-window Gabor frame.
{"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":"16 1","pages":""},"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}
Let $mathcal{M}$ and $mathcal{N}$ be nests on separable Hilbert space. If the two nest algebras are distance less than 1 ($d(mathcal{T}(mathcal{M}),mathcal{T}(mathcal{N})) < 1$), then the nests are distance less than 1 ($d(mathcal{M},mathcal{N})<1$). If the nests are distance less than 1 apart, then the nest algebras are similar, i.e. there is an invertible $S$ such that $Smathcal{M} = mathcal{N}$, so that $S mathcal{T}(mathcal{M})S^{-1} = mathcal{T}(mathcal{N})$. However there are examples of nests closer than 1 for which the nest algebras are distance 1 apart.
{"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":"1 1","pages":""},"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}
We investigate groups that act amenably on their Higson corona (also known as bi-exact groups) and we provide reformulations of this in relation to the stable Higson corona, nuclearity of crossed products and to positive type kernels. We further investigate implications of this in relation to the Baum-Connes conjecture, and prove that Gromov hyperbolic groups have isomorphic equivariant K-theories of their Gromov boundary and their stable Higson corona.
我们研究了可作用于其希格森冕的群组（也称为双作用群组），并结合稳定希格森冕、交叉积的核性和正型核对此进行了重新阐述。我们进一步研究了这一点与鲍姆-康恩猜想（Baum-Connes conjecture）之间的关系，并证明了格罗莫夫双曲群的格罗莫夫边界和稳定希格森冕具有同构向量 K 理论。
{"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":"104 1","pages":""},"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}
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