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Shubin calculi for actions of graded Lie groups 梯度李群作用的舒宾计算
arXiv - MATH - Operator Algebras Pub Date : 2024-07-19 DOI: arxiv-2407.14347
Eske Ewert, Philipp Schmitt
{"title":"Shubin calculi for actions of graded Lie groups","authors":"Eske Ewert, Philipp Schmitt","doi":"arxiv-2407.14347","DOIUrl":"https://doi.org/arxiv-2407.14347","url":null,"abstract":"In this article, we develop a calculus of Shubin type pseudodifferential\u0000operators on certain non-compact spaces, using a groupoid approach similar to\u0000the one of van Erp and Yuncken. More concretely, we consider actions of graded\u0000Lie groups on graded vector spaces and study pseudodifferential operators that\u0000generalize fundamental vector fields and multiplication by polynomials. Our two\u0000main examples of elliptic operators in this calculus are Rockland operators\u0000with a potential and the generalizations of the harmonic oscillator to the\u0000Heisenberg group due to Rottensteiner-Ruzhansky. Deforming the action of the graded group, we define a tangent groupoid which\u0000connects pseudodifferential operators to their principal (co)symbols. We show\u0000that our operators form a calculus that is asymptotically complete. Elliptic\u0000elements in the calculus have parametrices, are hypoelliptic, and can be\u0000characterized in terms of a Rockland condition. Moreover, we study the mapping\u0000properties as well as the spectra of our operators on Sobolev spaces and\u0000compare our calculus to the Shubin calculus on $mathbb R^n$ and its\u0000anisotropic generalizations.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740680","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
General K-closedness results in noncommutative Lebesgue spaces and applications to the real interpolation of noncommutative martingale Hardy spaces 非交换勒贝格空间中的一般K封闭性结果及其在非交换马丁格哈代空间实插值中的应用
arXiv - MATH - Operator Algebras Pub Date : 2024-07-17 DOI: arxiv-2407.12335
Hugues Moyart
{"title":"General K-closedness results in noncommutative Lebesgue spaces and applications to the real interpolation of noncommutative martingale Hardy spaces","authors":"Hugues Moyart","doi":"arxiv-2407.12335","DOIUrl":"https://doi.org/arxiv-2407.12335","url":null,"abstract":"In this paper, we establish a new general $K$-closedness result in the\u0000context of real interpolation of noncommutative Lebesgue spaces involving\u0000filtrations. As an application, we derive $K$-closedness results for various\u0000classes of noncommutative martingale Hardy spaces, addressing a problem raised\u0000by Randrianantoanina. The proof of this general result adapts Bourgain's\u0000approach to the real interpolation of classical Hardy spaces on the disk within\u0000the framework of noncommutative martingales.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740682","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 Hastings factorization for quantum many-body systems in the infinite volume setting 论无限体积环境中量子多体系统的哈斯汀因式分解
arXiv - MATH - Operator Algebras Pub Date : 2024-07-17 DOI: arxiv-2407.12324
Ayumi Ukai
{"title":"On Hastings factorization for quantum many-body systems in the infinite volume setting","authors":"Ayumi Ukai","doi":"arxiv-2407.12324","DOIUrl":"https://doi.org/arxiv-2407.12324","url":null,"abstract":"We will give self-contained and detailed proofs of the (multidimensional)\u0000Hastings factorization as well as a (1-dimensional) area law in a wider setup\u0000than previous works. Especially, they are applicable to both quantum spin and\u0000fermion chains of infinite volume as special cases.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745948","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
Strictly outer actions of locally compact groups: beyond the full factor case 局部紧凑群的严格外作用：超越全因子情况
arXiv - MATH - Operator Algebras Pub Date : 2024-07-16 DOI: arxiv-2407.11738
Basile Morando
{"title":"Strictly outer actions of locally compact groups: beyond the full factor case","authors":"Basile Morando","doi":"arxiv-2407.11738","DOIUrl":"https://doi.org/arxiv-2407.11738","url":null,"abstract":"We show that, given a continuous action $alpha$ of a locally compact group\u0000$G$ on a factor $M$, the relative commutant $M'cap(Mrtimes_{alpha} G)$ is\u0000contained in $Mrtimes_{alpha} H$ where $H$ is the subgroup of elements acting\u0000without spectral gap. As a corollary, we answer a question of Marrakchi and\u0000Vaes by showing that if $M$ is semifinite and $alpha_g$ is not approximately\u0000inner for all $gneq 1$, then $M'cap (Mrtimes_{alpha} G)=mathbb{C}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717871","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
Complete Metric Approximation Property For Mixed $q$-Deformed Araki-Woods Factors 混合 q$ 变形荒木因子的完整公制近似性质
arXiv - MATH - Operator Algebras Pub Date : 2024-07-15 DOI: arxiv-2407.10619
Panchugopal Bikram, Rajeeb Mohanta, Kunal Krishna Mukherjee
{"title":"Complete Metric Approximation Property For Mixed $q$-Deformed Araki-Woods Factors","authors":"Panchugopal Bikram, Rajeeb Mohanta, Kunal Krishna Mukherjee","doi":"arxiv-2407.10619","DOIUrl":"https://doi.org/arxiv-2407.10619","url":null,"abstract":"The main result of this paper is to establish the $text{weak}^*$ completely\u0000metric approximation property ($w^*$-CMAP) for the mixed $q$-deformed\u0000Araki-Woods factors for all symmetric matrices with entries $-1< q_{i,j}< 1$,\u0000using an ultraproduct embedding of the mixed $q$-deformed Araki-Woods factors.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717873","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
Noncommutative topological boundaries and amenable invariant random intermediate subalgebras 非交换拓扑边界和可变随机中间子代数
arXiv - MATH - Operator Algebras Pub Date : 2024-07-15 DOI: arxiv-2407.10905
Shuoxing Zhou
{"title":"Noncommutative topological boundaries and amenable invariant random intermediate subalgebras","authors":"Shuoxing Zhou","doi":"arxiv-2407.10905","DOIUrl":"https://doi.org/arxiv-2407.10905","url":null,"abstract":"As an analogue of topological boundary of discrete groups $Gamma$, we define\u0000the noncommutative topological boundary of tracial von Neumann algebras\u0000$(M,tau)$ and apply it to generalize the main results of [AHO23], showing that\u0000for a trace preserving action $Gamma curvearrowright(A,tau_A)$ on an\u0000amenable tracial von Neumann algebra, a $Gamma$-invariant measure\u0000$muinmathrm{Prob}(mathrm{SA}(Gammaltimes A))$ supported on amenable\u0000intermediate subalgebras between $A$ and $Gammaltimes A$ is necessary\u0000supported on the subalgebras of $mathrm{Rad}(Gamma)ltimes A$. By taking\u0000$(A,tau)=L^infty(X,nu_X)$ for a free p.m.p. action $Gamma\u0000curvearrowright(X,nu_X)$, we obtain a similar results for the invariant\u0000random subequivalence relations of $mathcal{R}_{Gamma curvearrowright X}$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717872","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
Ternary rings of operators and their linking von Neumann algebras 三元算子环及其关联冯-诺依曼代数
arXiv - MATH - Operator Algebras Pub Date : 2024-07-14 DOI: arxiv-2407.10154
Liguang Wang, Ngai-Ching Wong
{"title":"Ternary rings of operators and their linking von Neumann algebras","authors":"Liguang Wang, Ngai-Ching Wong","doi":"arxiv-2407.10154","DOIUrl":"https://doi.org/arxiv-2407.10154","url":null,"abstract":"In this short note, we show that a von Neumann algebra can be written as the\u0000linking von Neumann algebra of a $W^ast$-ternary ring of operators\u0000($W^ast$-TRO, in short), if and only if, it contains no abelian direct\u0000summand. We also provide some new characterizations for nuclear TROs and\u0000$W^ast$-exact TROs.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717874","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
Operational 2-local automorphisms/derivations 操作性 2-局部自动形态/派生
arXiv - MATH - Operator Algebras Pub Date : 2024-07-14 DOI: arxiv-2407.10150
Liguang Wang, Ngai-Ching Wong
{"title":"Operational 2-local automorphisms/derivations","authors":"Liguang Wang, Ngai-Ching Wong","doi":"arxiv-2407.10150","DOIUrl":"https://doi.org/arxiv-2407.10150","url":null,"abstract":"Let $phi: Ato A$ be a (not necessarily linear, additive or continuous) map of a standard operator algebra. Suppose for any $a,bin A$ there is an\u0000algebra automorphism $theta_{a,b}$ of $ A$ such that begin{align*}\u0000phi(a)phi(b) = theta_{a,b}(ab). end{align*} We show that either $phi$ or\u0000$-phi$ is a linear Jordan homomorphism. Similar results are obtained when any of the following conditions is\u0000satisfied: begin{align*} phi(a) + phi(b) &= theta_{a,b}(a+b), \u0000phi(a)phi(b)+phi(b)phi(a) &= theta_{a,b}(ab+ba), quadtext{or} \u0000phi(a)phi(b)phi(a) &= theta_{a,b}(aba). end{align*} We also show that a map $phi: Mto M$ of a semi-finite von Neumann algebra $\u0000M$ is a linear derivation if for every $a,bin M$ there is a linear derivation\u0000$D_{a,b}$ of $M$ such that $$ phi(a)b + aphi(b) = D_{a,b}(ab). $$","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717875","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 closure operations in the space of subgroups and applications 子群空间中的闭合运算及其应用
arXiv - MATH - Operator Algebras Pub Date : 2024-07-14 DOI: arxiv-2407.10222
Dominik Francoeur, Adrien Le Boudec
{"title":"On closure operations in the space of subgroups and applications","authors":"Dominik Francoeur, Adrien Le Boudec","doi":"arxiv-2407.10222","DOIUrl":"https://doi.org/arxiv-2407.10222","url":null,"abstract":"We establish some interactions between uniformly recurrent subgroups (URSs)\u0000of a group $G$ and cosets topologies $tau_mathcal{N}$ on $G$ associated to a\u0000family $mathcal{N}$ of normal subgroups of $G$. We show that when\u0000$mathcal{N}$ consists of finite index subgroups of $G$, there is a natural\u0000closure operation $mathcal{H} mapsto mathrm{cl}_mathcal{N}(mathcal{H})$\u0000that associates to a URS $mathcal{H}$ another URS\u0000$mathrm{cl}_mathcal{N}(mathcal{H})$, called the $tau_mathcal{N}$-closure\u0000of $mathcal{H}$. We give a characterization of the URSs $mathcal{H}$ that are\u0000$tau_mathcal{N}$-closed in terms of stabilizer URSs. This has consequences on\u0000arbitrary URSs when $G$ belongs to the class of groups for which every faithful\u0000minimal profinite action is topologically free. We also consider the largest\u0000amenable URS $mathcal{A}_G$, and prove that for certain coset topologies on\u0000$G$, almost all subgroups $H in mathcal{A}_G$ have the same closure. For\u0000groups in which amenability is detected by a set of laws, we deduce a criterion\u0000for $mathcal{A}_G$ to be a singleton based on residual properties of $G$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717878","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
Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property 具有弱三元罗克林性质的有限群作用交叉积的稳定秩
arXiv - MATH - Operator Algebras Pub Date : 2024-07-13 DOI: arxiv-2407.09867
Xiaochun Fang, Zhongli Wang
{"title":"Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property","authors":"Xiaochun Fang, Zhongli Wang","doi":"arxiv-2407.09867","DOIUrl":"https://doi.org/arxiv-2407.09867","url":null,"abstract":"Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra,\u0000let $G$ be a finite group, and let $alphacolon Grightarrow mathrm{Aut}(A)$\u0000be an action of $G$ on $A$ which has the weak tracial Rokhlin property. We\u0000prove that if $A$ has property (TM), then the crossed product $Artimes_alpha\u0000G$ has property (TM). As a corollary, if $A$ is an infinite-dimensional\u0000separable simple unital C*-algebra which has stable rank one and strict\u0000comparison, $alphacolon Grightarrow mathrm{Aut}(A)$ is an action of a\u0000finite group $G$ on $A$ with the weak tracial Rokhlin property, then\u0000$Artimes_alpha G$ has stable rank one.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717876","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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