In this article, we develop a calculus of Shubin type pseudodifferential operators on certain non-compact spaces, using a groupoid approach similar to the one of van Erp and Yuncken. More concretely, we consider actions of graded Lie groups on graded vector spaces and study pseudodifferential operators that generalize fundamental vector fields and multiplication by polynomials. Our two main examples of elliptic operators in this calculus are Rockland operators with a potential and the generalizations of the harmonic oscillator to the Heisenberg group due to Rottensteiner-Ruzhansky. Deforming the action of the graded group, we define a tangent groupoid which connects pseudodifferential operators to their principal (co)symbols. We show that our operators form a calculus that is asymptotically complete. Elliptic elements in the calculus have parametrices, are hypoelliptic, and can be characterized in terms of a Rockland condition. Moreover, we study the mapping properties as well as the spectra of our operators on Sobolev spaces and compare our calculus to the Shubin calculus on $mathbb R^n$ and its anisotropic generalizations.
在这篇文章中，我们使用与 van Erp 和 Yuncken 类似的类群方法，在某些非紧凑空间上建立了舒宾型伪微分算子的微积分。更具体地说，我们考虑了分级李群在分级向量空间上的作用，并研究了泛化基本向量场和多项式乘法的伪微分算子。在这种微积分中，我们的椭圆算子的双域例子是具有势的罗克兰算子和罗滕斯泰纳-鲁赞斯基对海森堡群的谐振子的泛化。我们对有级群的作用进行变形，定义了一个切线群，它将伪微分算子与其主（共）符号连接起来。我们证明，我们的算子构成了一个渐近完备的微积分。微积分中的椭圆元素具有参数，是次椭圆的，可以用洛克兰条件来描述。此外，我们还研究了我们的算子在索波列夫空间上的映射性质和谱，并将我们的微积分与 $mathbb R^n$ 上的舒宾微积分及其各向异性广义进行了比较。
{"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":"63 1","pages":""},"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}
In this paper, we establish a new general $K$-closedness result in the context of real interpolation of noncommutative Lebesgue spaces involving filtrations. As an application, we derive $K$-closedness results for various classes of noncommutative martingale Hardy spaces, addressing a problem raised by Randrianantoanina. The proof of this general result adapts Bourgain's approach to the real interpolation of classical Hardy spaces on the disk within the framework of noncommutative martingales.
{"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":"63 1","pages":""},"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}
We will give self-contained and detailed proofs of the (multidimensional) Hastings factorization as well as a (1-dimensional) area law in a wider setup than previous works. Especially, they are applicable to both quantum spin and fermion chains of infinite volume as special cases.
{"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":"40 1","pages":""},"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}
We show that, given a continuous action $alpha$ of a locally compact group $G$ on a factor $M$, the relative commutant $M'cap(Mrtimes_{alpha} G)$ is contained in $Mrtimes_{alpha} H$ where $H$ is the subgroup of elements acting without spectral gap. As a corollary, we answer a question of Marrakchi and Vaes by showing that if $M$ is semifinite and $alpha_g$ is not approximately inner for all $gneq 1$, then $M'cap (Mrtimes_{alpha} G)=mathbb{C}$.
{"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":"25 1","pages":""},"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}
The main result of this paper is to establish the $text{weak}^*$ completely metric approximation property ($w^*$-CMAP) for the mixed $q$-deformed Araki-Woods factors for all symmetric matrices with entries $-1< q_{i,j}< 1$, using an ultraproduct embedding of the mixed $q$-deformed Araki-Woods factors.
{"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":"30 1","pages":""},"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}
As an analogue of topological boundary of discrete groups $Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M,tau)$ and apply it to generalize the main results of [AHO23], showing that for a trace preserving action $Gamma curvearrowright(A,tau_A)$ on an amenable tracial von Neumann algebra, a $Gamma$-invariant measure $muinmathrm{Prob}(mathrm{SA}(Gammaltimes A))$ supported on amenable intermediate subalgebras between $A$ and $Gammaltimes A$ is necessary supported on the subalgebras of $mathrm{Rad}(Gamma)ltimes A$. By taking $(A,tau)=L^infty(X,nu_X)$ for a free p.m.p. action $Gamma curvearrowright(X,nu_X)$, we obtain a similar results for the invariant random subequivalence relations of $mathcal{R}_{Gamma curvearrowright X}$.
{"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":"11 1","pages":""},"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}
In this short note, we show that a von Neumann algebra can be written as the linking von Neumann algebra of a $W^ast$-ternary ring of operators ($W^ast$-TRO, in short), if and only if, it contains no abelian direct summand. We also provide some new characterizations for nuclear TROs and $W^ast$-exact TROs.
在这篇短文中，我们证明了当且仅当一个冯-诺依曼代数不包含无邻直和时，它可以被写成一个 $W^ast$-ternary ring of operators（简称 $W^ast$-TRO）的连接冯-诺依曼代数。我们还为核TRO和$W^ast$-Exact TRO提供了一些新的特征。
{"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":"49 1","pages":""},"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}
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 algebra automorphism $theta_{a,b}$ of $ A$ such that begin{align*} phi(a)phi(b) = theta_{a,b}(ab). end{align*} We show that either $phi$ or $-phi$ is a linear Jordan homomorphism. Similar results are obtained when any of the following conditions is satisfied: begin{align*} phi(a) + phi(b) &= theta_{a,b}(a+b), phi(a)phi(b)+phi(b)phi(a) &= theta_{a,b}(ab+ba), quadtext{or} phi(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 $ M$ is a linear derivation if for every $a,bin M$ there is a linear derivation $D_{a,b}$ of $M$ such that $$ phi(a)b + aphi(b) = D_{a,b}(ab). $$
{"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":"59 1","pages":""},"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}
We establish some interactions between uniformly recurrent subgroups (URSs) of a group $G$ and cosets topologies $tau_mathcal{N}$ on $G$ associated to a family $mathcal{N}$ of normal subgroups of $G$. We show that when $mathcal{N}$ consists of finite index subgroups of $G$, there is a natural closure operation $mathcal{H} mapsto mathrm{cl}_mathcal{N}(mathcal{H})$ that associates to a URS $mathcal{H}$ another URS $mathrm{cl}_mathcal{N}(mathcal{H})$, called the $tau_mathcal{N}$-closure of $mathcal{H}$. We give a characterization of the URSs $mathcal{H}$ that are $tau_mathcal{N}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when $G$ belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $mathcal{A}_G$, and prove that for certain coset topologies on $G$, almost all subgroups $H in mathcal{A}_G$ have the same closure. For groups in which amenability is detected by a set of laws, we deduce a criterion for $mathcal{A}_G$ to be a singleton based on residual properties of $G$.
{"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":"63 1","pages":""},"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}
Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, let $G$ be a finite group, and let $alphacolon Grightarrow mathrm{Aut}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. We prove that if $A$ has property (TM), then the crossed product $Artimes_alpha G$ has property (TM). As a corollary, if $A$ is an infinite-dimensional separable simple unital C*-algebra which has stable rank one and strict comparison, $alphacolon Grightarrow mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then $Artimes_alpha G$ has stable rank one.
{"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":"49 1","pages":""},"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}
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