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Universal covering groups of unitary groups of von Neumann algebras 冯-诺依曼代数单元群的普遍覆盖群
arXiv - MATH - Operator Algebras Pub Date : 2024-08-25 DOI: arxiv-2408.13710
Pawel Sarkowicz
{"title":"Universal covering groups of unitary groups of von Neumann algebras","authors":"Pawel Sarkowicz","doi":"arxiv-2408.13710","DOIUrl":"https://doi.org/arxiv-2408.13710","url":null,"abstract":"We give a short and simple proof, utilizing the pre-determinant of P. de la\u0000Harpe and G. Skandalis, that the universal covering group of the unitary group\u0000of a II$_1$ von Neumann algebra $mathcal{M}$, when equipped with the norm\u0000topology, splits algebraically as the direct product of the self-adjoint part\u0000of its center and the unitary group $U(mathcal{M})$. Thus, when $mathcal{M}$\u0000is a II$_1$ factor, the universal covering group of $U(mathcal{M})$ is\u0000algebraically isomorphic to the direct product $mathbb{R} times\u0000U(mathcal{M})$. In particular, the question of P. de la Harpe and D. McDuff of\u0000whether the universal cover of $U(mathcal{M})$ is a perfect group is answered\u0000in the negative.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195043","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 Khintchine inequality for central Fourier series on non-Kac compact quantum groups 非 Kac 紧凑量子群上中心傅里叶级数的 Khintchine 不等式
arXiv - MATH - Operator Algebras Pub Date : 2024-08-24 DOI: arxiv-2408.13519
Sang-Gyun Youn
{"title":"A Khintchine inequality for central Fourier series on non-Kac compact quantum groups","authors":"Sang-Gyun Youn","doi":"arxiv-2408.13519","DOIUrl":"https://doi.org/arxiv-2408.13519","url":null,"abstract":"The study of Khintchin inequalities has a long history in abstract harmonic\u0000analysis. While there is almost no possibility of non-trivial Khintchine\u0000inequality for central Fourier series on compact connected semisimple Lie\u0000groups, we demonstrate a strong contrast within the framework of compact\u0000quantum groups. Specifically, we establish a Khintchine inequality with\u0000operator coefficients for arbitrary central Fourier series in a large class of\u0000non-Kac compact quantum groups. The main examples include the Drinfeld-Jimbo\u0000$q$-deformations $G_q$, the free orthogonal quantum groups $O_F^+$, and the\u0000quantum automorphism group $G_{aut}(B,psi)$ with a $delta$-form $psi$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195042","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 inter-relationship between isomorphisms of commutative and isomorphisms of non-commutative $log$-algebras 交换代数的同构与非交换代数的同构之间的相互关系
arXiv - MATH - Operator Algebras Pub Date : 2024-08-24 DOI: arxiv-2408.13527
Rustam Abdullaev, Azizkhon Azizov
{"title":"The inter-relationship between isomorphisms of commutative and isomorphisms of non-commutative $log$-algebras","authors":"Rustam Abdullaev, Azizkhon Azizov","doi":"arxiv-2408.13527","DOIUrl":"https://doi.org/arxiv-2408.13527","url":null,"abstract":"This paper establishes a necessary and sufficient condition for the\u0000coincidence of non-commutative $log$-algebras constructed from different exact\u0000normal semifinite traces. Consequently, we provide a criterion for the\u0000isomorphism of $log$-algebras built on non-commutative von Neumann algebras\u0000with different exact normal semifinite traces. Additionally, we demonstrate a\u0000connection between the isomorphism of non-commutative $log$-algebras and the\u0000isomorphism of the corresponding $log$-algebras constructed on the center of\u0000these von Neumann algebras. Furthermore, we present a necessary and sufficient\u0000condition for the isomorphism of $log$-algebras derived from different von\u0000Neumann algebras of type $I_n$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195045","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
Conditional representation stability, classification of $*$-homomorphisms, and eta invariants 条件表示稳定性、$*$同构的分类和 eta 不变量
arXiv - MATH - Operator Algebras Pub Date : 2024-08-23 DOI: arxiv-2408.13350
Rufus Willett
{"title":"Conditional representation stability, classification of $*$-homomorphisms, and eta invariants","authors":"Rufus Willett","doi":"arxiv-2408.13350","DOIUrl":"https://doi.org/arxiv-2408.13350","url":null,"abstract":"A quasi-representation of a group is a map from the group into a matrix\u0000algebra (or similar object) that approximately satisfies the relations needed\u0000to be a representation. Work of many people starting with Kazhdan and\u0000Voiculescu, and recently advanced by Dadarlat, Eilers-Shulman-So{}rensen and\u0000others, has shown that there are topological obstructions to approximating\u0000unitary quasi-representations of groups by honest representations, where\u0000`approximation' is understood to be with respect to the operator norm. The purpose of this paper is to explore whether approximation is possible if\u0000the known obstructions vanish, partially generalizing work of Gong-Lin and\u0000Eilers-Loring-Pedersen for the free abelian group of rank two, and the Klein\u0000bottle group. We show that this is possible, at least in a weak sense, for some\u0000`low-dimensional' groups including fundamental groups of closed surfaces,\u0000certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental\u0000groups of three manifolds. The techniques used in the paper are $K$-theoretic: they have their origin in\u0000Baum-Connes-Kasparov type assembly maps, and in the Elliott program to classify\u0000$C^*$-algebras; Kasparov's bivariant KK-theory is a crucial tool. The key new\u0000technical ingredients are: a stable uniqueness theorem in the sense of\u0000Dadarlat-Eilers and Lin that works for non-exact $C^*$-algebras; and an\u0000analysis of maps on $K$-theory with finite coefficients in terms of the\u0000relative eta invariants of Atiyah-Patodi-Singer. Despite the proofs going\u0000through $K$-theoretic machinery, the main theorems can be stated in elementary\u0000terms that do not need any $K$-theory.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195048","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
Toward stringy horizons 走向多线地平线
arXiv - MATH - Operator Algebras Pub Date : 2024-08-22 DOI: arxiv-2408.12642
Elliott Gesteau, Hong Liu
{"title":"Toward stringy horizons","authors":"Elliott Gesteau, Hong Liu","doi":"arxiv-2408.12642","DOIUrl":"https://doi.org/arxiv-2408.12642","url":null,"abstract":"We take a first step towards developing a new language to describe causal\u0000structure, event horizons, and quantum extremal surfaces (QES) for the bulk\u0000description of holographic systems beyond the standard Einstein gravity regime.\u0000By considering the structure of boundary operator algebras, we introduce a\u0000stringy ``causal depth parameter'', which quantifies the depth of the emergent\u0000radial direction in the bulk, and a certain notion of ergodicity on the\u0000boundary. We define stringy event horizons in terms of the half-sided inclusion\u0000property, which is related to a stronger notion of boundary ergodic or quantum\u0000chaotic behavior. Using our definition, we argue that above the Hawking--Page\u0000temperature, there is an emergent sharp horizon structure in the large $N$\u0000limit of $mathcal{N}=4$ Super-Yang--Mills at finite nonzero 't Hooft coupling.\u0000In contrast, some previously considered toy models of black hole information\u0000loss do not have a stringy horizon. Our methods can also be used to probe\u0000violations of the equivalence principle for the bulk gravitational system, and\u0000to explore aspects of stringy nonlocality.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195047","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
Hermitian crossed product Banach algebras 赫米特交叉积巴拿赫代数
arXiv - MATH - Operator Algebras Pub Date : 2024-08-21 DOI: arxiv-2408.11466
Rachid El Harti, Paulo R. Pinto
{"title":"Hermitian crossed product Banach algebras","authors":"Rachid El Harti, Paulo R. Pinto","doi":"arxiv-2408.11466","DOIUrl":"https://doi.org/arxiv-2408.11466","url":null,"abstract":"We show that the Banach *-algebra $ell^1(G,A,alpha)$, arising from a\u0000C*-dynamical system $(A,G,alpha)$, is an hermitian Banach algebra if the\u0000discrete group $G$ is finite or abelian (or more generally, a finite extension\u0000of a nilpotent group). As a corollary, we obtain that $ell^1(mathbb{Z},C(X),alpha)$ is hermitian,\u0000for every topological dynamical system $Sigma = (X, sigma)$, where $sigma:\u0000Xto X$ is a homeomorphism of a compact Hausdorff space $X$ and the action is\u0000$alpha_n(f)=fcirc sigma^{-n}$ with $ninmathbb{Z}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195052","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
Minimal covers with continuity-preserving transfer operators for topological dynamical systems 拓扑动力系统的最小盖与连续性保持转移算子
arXiv - MATH - Operator Algebras Pub Date : 2024-08-21 DOI: arxiv-2408.11917
Kevin Aguyar Brix, Jeremy B. Hume, Xin Li
{"title":"Minimal covers with continuity-preserving transfer operators for topological dynamical systems","authors":"Kevin Aguyar Brix, Jeremy B. Hume, Xin Li","doi":"arxiv-2408.11917","DOIUrl":"https://doi.org/arxiv-2408.11917","url":null,"abstract":"Given a non-invertible dynamical system with a transfer operator, we show\u0000there is a minimal cover with a transfer operator that preserves continuous\u0000functions. We also introduce an essential cover with even stronger continuity\u0000properties. For one-sided sofic subshifts, this generalizes the Krieger and\u0000Fischer covers, respectively. Our construction is functorial in the sense that\u0000certain equivariant maps between dynamical systems lift to equivariant maps\u0000between their covers, and these maps also satisfy better regularity properties.\u0000As applications, we identify finiteness conditions which ensure that the\u0000thermodynamic formalism is valid for the covers. This establishes the\u0000thermodynamic formalism for a large class of non-invertible dynamical systems,\u0000e.g. certain piecewise invertible maps. When applied to semi-'etale groupoids,\u0000our minimal covers produce 'etale groupoids which are models for\u0000$C^*$-algebras constructed by Thomsen. The dynamical covers and groupoid covers\u0000are unified under the common framework of topological graphs.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195051","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 soficity for certain fundamental groups of graphs of groups 关于群的图的某些基群的soficity
arXiv - MATH - Operator Algebras Pub Date : 2024-08-21 DOI: arxiv-2408.11724
David Gao, Srivatsav Kunnawalkam Elayavalli, Mahan Mj
{"title":"On soficity for certain fundamental groups of graphs of groups","authors":"David Gao, Srivatsav Kunnawalkam Elayavalli, Mahan Mj","doi":"arxiv-2408.11724","DOIUrl":"https://doi.org/arxiv-2408.11724","url":null,"abstract":"In this note we study a family of graphs of groups over arbitrary base graphs\u0000where all vertex groups are isomorphic to a fixed countable sofic group $G$,\u0000and all edge groups $H<G$ are such that the embeddings of $H$ into $G$ are\u0000identical everywhere. We prove soficity for this family of groups under a\u0000flexible technical hypothesis for $H$ called $sigma$-co-sofic. This proves\u0000soficity for group doubles $*_H G$, where $H<G$ is an arbitrary separable\u0000subgroup and $G$ is countable and sofic. This includes arbitrary finite index\u0000group doubles of sofic groups among various other examples.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195054","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
Remark on height functions 高度函数备注
arXiv - MATH - Operator Algebras Pub Date : 2024-08-21 DOI: arxiv-2408.12020
Igor V. Nikolaev
{"title":"Remark on height functions","authors":"Igor V. Nikolaev","doi":"arxiv-2408.12020","DOIUrl":"https://doi.org/arxiv-2408.12020","url":null,"abstract":"Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety\u0000over $k$. We use the $K$-theory of a $C^*$-algebra $A_V$ associated to $V(k)$\u0000to define a height of points of $V(k)$. The corresponding counting function is\u0000calculated and we show that it coincides with the known formulas for $n=1$. As\u0000an application, it is proved that the set $V(k)$ is finite, whenever the sum of\u0000the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the\u0000$n$-dimensional Minkowski question-mark function studied by Panti and others.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195049","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 chromatic numbers of products of quantum graphs 量子图积的量子色度数
arXiv - MATH - Operator Algebras Pub Date : 2024-08-21 DOI: arxiv-2408.11911
Rolando de Santiago, A. Meenakshi McNamara
{"title":"Quantum chromatic numbers of products of quantum graphs","authors":"Rolando de Santiago, A. Meenakshi McNamara","doi":"arxiv-2408.11911","DOIUrl":"https://doi.org/arxiv-2408.11911","url":null,"abstract":"We define the Cartesian, Categorical, and Lexicographic, and Strong products\u0000of quantum graphs. We provide bounds on the quantum chromatic number of these\u0000products in terms of the quantum chromatic number of the factors. To adequately\u0000describe bounds on the lexicographic product of quantum graphs, we provide a\u0000notion of a quantum $b$-fold chromatic number for quantum graphs.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195046","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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