We study the dynamics of non-relativistic fermions in $mathbb R^d$ interacting through a pair potential. Employing methods developed by Buchholz in the framework of resolvent algebras, we identify an extension of the CAR algebra where the dynamics acts as a group of *-automorphisms, which are continuous in time in all sectors for fixed particle numbers. In addition, we identify a suitable dense subalgebra where the time evolution is also strongly continuous. Finally, we briefly discuss how this framework could be used to construct KMS states in the future.
{"title":"On the thermodynamic limit of interacting fermions in the continuum","authors":"Oliver Siebert","doi":"arxiv-2409.10495","DOIUrl":"https://doi.org/arxiv-2409.10495","url":null,"abstract":"We study the dynamics of non-relativistic fermions in $mathbb R^d$\u0000interacting through a pair potential. Employing methods developed by Buchholz\u0000in the framework of resolvent algebras, we identify an extension of the CAR\u0000algebra where the dynamics acts as a group of *-automorphisms, which are\u0000continuous in time in all sectors for fixed particle numbers. In addition, we\u0000identify a suitable dense subalgebra where the time evolution is also strongly\u0000continuous. Finally, we briefly discuss how this framework could be used to\u0000construct KMS states in the future.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259806","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 article we consider the generalized integral operators acting on the Hilbert space $H^2$. We characterize when these operators are uniform, strong and weakly asymptotic Toeplitz and Hankel operators. Moreover we completely describe the symbols $g$ for which these operators are essentially Hankel and essentially Toeplitz.
{"title":"On asymptotic and essential Toeplitz and Hankel integral operator","authors":"C. Bellavita, G. Stylogiannis","doi":"arxiv-2409.10014","DOIUrl":"https://doi.org/arxiv-2409.10014","url":null,"abstract":"In this article we consider the generalized integral operators acting on the\u0000Hilbert space $H^2$. We characterize when these operators are uniform, strong\u0000and weakly asymptotic Toeplitz and Hankel operators. Moreover we completely\u0000describe the symbols $g$ for which these operators are essentially Hankel and\u0000essentially Toeplitz.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259808","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 notion of Shilov boundary ideal for a local operator system and investigate some its properties.
本文介绍了局部算子系统的 Shilov 边界理想概念，并研究了它的一些性质。
{"title":"The Shilov boundary for a local operator system","authors":"Maria Joiţa","doi":"arxiv-2409.10474","DOIUrl":"https://doi.org/arxiv-2409.10474","url":null,"abstract":"In this paper, we introduce the notion of Shilov boundary ideal for a local\u0000operator system and investigate some its properties.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259809","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 a simple and elementary proof that the tracial state space of a unital C$^*$-algebra is a Choquet simplex, using the center-valued trace on a finite von Neumann algebra.
我们利用无穷 von Neumann 代数上的中心值迹，给出了一个简单而基本的证明，即无穷 C$^*$ 代数的三态空间是一个 Choquet 单纯形。
{"title":"The Space of Tracial States on a C$^*$-Algebra","authors":"Bruce Blackadar, Mikael Rørdam","doi":"arxiv-2409.09644","DOIUrl":"https://doi.org/arxiv-2409.09644","url":null,"abstract":"We give a simple and elementary proof that the tracial state space of a\u0000unital C$^*$-algebra is a Choquet simplex, using the center-valued trace on a\u0000finite von Neumann algebra.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259805","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}
Based on his claims in 1990, Rosenberg conjectured in 1997 that the negative algebraic $K$-groups of C*-algebras are invariant under continuous homotopy. Contrary to his expectation, we prove that such invariance holds for $K_{-1}$ of arbitrary Banach rings by establishing a certain continuity result. We also construct examples demonstrating that similar continuity results do not hold for lower $K$-groups.
{"title":"Rosenberg's conjecture for the first negative $K$-group","authors":"Ko Aoki","doi":"arxiv-2409.09651","DOIUrl":"https://doi.org/arxiv-2409.09651","url":null,"abstract":"Based on his claims in 1990, Rosenberg conjectured in 1997 that the negative\u0000algebraic $K$-groups of C*-algebras are invariant under continuous homotopy.\u0000Contrary to his expectation, we prove that such invariance holds for $K_{-1}$\u0000of arbitrary Banach rings by establishing a certain continuity result. We also\u0000construct examples demonstrating that similar continuity results do not hold\u0000for lower $K$-groups.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259810","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 introduce the extension graph of graph product of groups and study its geometry. This enables us to study properties of graph products of groups by exploiting large scale geometry of its defining graph. In particular, we show that the extension graph exhibits the same phenomenon about asymptotic dimension as quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. Moreover, we present three applications of the extension graph of graph product when a defining graph is hyperbolic. First, we provide a new class of convergence groups by considering the action of graph product of finite groups on a compactification of the extension graph and identify the if and only if condition for this action to be geometrically finite. Secondly, we prove relative hyperbolicity of the semi-direct product of groups that interpolates between wreath product and free product. Finally, we provide a new class of graph product of finite groups whose group von Neumnann algebra is strongly solid.
我们介绍了群的图积的扩展图，并研究了它的几何学。这使我们能够通过利用其定义图的大尺度几何来研究群的图积的性质。特别是，我们证明了扩展图与贝斯特维纳-布罗姆伯格-藤原所研究的度量空间的准树状图在渐近维度上表现出相同的现象。此外，我们还介绍了当定义图为双曲图时图积的扩展图的三个应用。首先，我们通过考虑有限群的图积对扩展图紧凑化的作用，提供了一类新的收敛群，并确定了该作用在几何上是无限的 "如果 "和 "唯一 "条件。其次，我们证明了介于花环积和自由积之间的群的半直接积的相对双曲性。最后，我们提供了一类新的有限群的图积，其群 von Neumnannalgebra 是强固的。
{"title":"Geometry and dynamics of the extension graph of graph product of groups","authors":"Koichi Oyakawa","doi":"arxiv-2409.09527","DOIUrl":"https://doi.org/arxiv-2409.09527","url":null,"abstract":"We introduce the extension graph of graph product of groups and study its\u0000geometry. This enables us to study properties of graph products of groups by\u0000exploiting large scale geometry of its defining graph. In particular, we show\u0000that the extension graph exhibits the same phenomenon about asymptotic\u0000dimension as quasi-trees of metric spaces studied by\u0000Bestvina-Bromberg-Fujiwara. Moreover, we present three applications of the\u0000extension graph of graph product when a defining graph is hyperbolic. First, we\u0000provide a new class of convergence groups by considering the action of graph\u0000product of finite groups on a compactification of the extension graph and\u0000identify the if and only if condition for this action to be geometrically\u0000finite. Secondly, we prove relative hyperbolicity of the semi-direct product of\u0000groups that interpolates between wreath product and free product. Finally, we\u0000provide a new class of graph product of finite groups whose group von Neumnann\u0000algebra is strongly solid.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259807","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 aim of this paper is to bridge noncommutative geometry with classical harmonic analysis on Banach spaces, focusing primarily on both classical and noncommutative $mathrm{L}^p$ spaces. Introducing a notion of Banach Fredholm module, we define new abelian groups, $mathrm{K}^{0}(mathcal{A},mathscr{B})$ and $mathrm{K}^{1}(mathcal{A},mathscr{B})$, of $mathrm{K}$-homology associated with an algebra $mathcal{A}$ and a suitable class $mathscr{B}$ of Banach spaces, such as the class of $mathrm{L}^p$-spaces. We establish index pairings of these groups with the $mathrm{K}$-theory groups of the algebra $mathcal{A}$. Subsequently, by considering (noncommutative) Hardy spaces, we uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm modules and culminating in index theorems. Moreover, by associating each reasonable sub-Markovian semigroup with a <>, we explain how this leads to (possibly kernel-degenerate) Banach Fredholm modules, thereby revealing the role of vectorial Riesz transforms in this context. Overall, our approach significantly integrates the analysis of operators on $mathrm{L}^p$-spaces into the expansive framework of noncommutative geometry, offering new perspectives.
{"title":"Classical harmonic analysis viewed through the prism of noncommutative geometry","authors":"Cédric Arhancet","doi":"arxiv-2409.07750","DOIUrl":"https://doi.org/arxiv-2409.07750","url":null,"abstract":"The aim of this paper is to bridge noncommutative geometry with classical\u0000harmonic analysis on Banach spaces, focusing primarily on both classical and\u0000noncommutative $mathrm{L}^p$ spaces. Introducing a notion of Banach Fredholm\u0000module, we define new abelian groups, $mathrm{K}^{0}(mathcal{A},mathscr{B})$\u0000and $mathrm{K}^{1}(mathcal{A},mathscr{B})$, of $mathrm{K}$-homology\u0000associated with an algebra $mathcal{A}$ and a suitable class $mathscr{B}$ of\u0000Banach spaces, such as the class of $mathrm{L}^p$-spaces. We establish index\u0000pairings of these groups with the $mathrm{K}$-theory groups of the algebra\u0000$mathcal{A}$. Subsequently, by considering (noncommutative) Hardy spaces, we\u0000uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm\u0000modules and culminating in index theorems. Moreover, by associating each\u0000reasonable sub-Markovian semigroup with a <<Banach noncommutative manifold>>,\u0000we explain how this leads to (possibly kernel-degenerate) Banach Fredholm\u0000modules, thereby revealing the role of vectorial Riesz transforms in this\u0000context. Overall, our approach significantly integrates the analysis of\u0000operators on $mathrm{L}^p$-spaces into the expansive framework of\u0000noncommutative geometry, offering new perspectives.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195014","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 survey a number of incompleteness results in operator algebras stemming from the recent undecidability result in quantum complexity theory known as $operatorname{MIP}^*=operatorname{RE}$, the most prominent of which is the G"odelian refutation of the Connes Embedding Problem. We also discuss the very recent use of $operatorname{MIP}^*=operatorname{RE}$ in refuting the Aldous-Lyons conjecture in probability theory.
{"title":"Undecidability and incompleteness in quantum information theory and operator algebras","authors":"Isaac Goldbring","doi":"arxiv-2409.08342","DOIUrl":"https://doi.org/arxiv-2409.08342","url":null,"abstract":"We survey a number of incompleteness results in operator algebras stemming\u0000from the recent undecidability result in quantum complexity theory known as\u0000$operatorname{MIP}^*=operatorname{RE}$, the most prominent of which is the\u0000G\"odelian refutation of the Connes Embedding Problem. We also discuss the very\u0000recent use of $operatorname{MIP}^*=operatorname{RE}$ in refuting the\u0000Aldous-Lyons conjecture in probability theory.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269306","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 $pin(1,infty)$. We show that there is an isomorphism from any separable unital subalgebra of $B(ell^{2})/K(ell^{2})$ onto a subalgebra of $B(ell^{p})/K(ell^{p})$ that preserves the Fredholm index. As a consequence, every separable $C^{*}$-algebra is isomorphic to a subalgebra of $B(ell^{p})/K(ell^{p})$. Another consequence is the existence of operators on $ell^{p}$ that behave like the essentially normal operators with arbitrary Fredholm indices in the Brown-Douglas-Fillmore theory.
{"title":"Embedding C*-algebras into the Calkin algebra of $ell^{p}$","authors":"March T. Boedihardjo","doi":"arxiv-2409.07386","DOIUrl":"https://doi.org/arxiv-2409.07386","url":null,"abstract":"Let $pin(1,infty)$. We show that there is an isomorphism from any separable\u0000unital subalgebra of $B(ell^{2})/K(ell^{2})$ onto a subalgebra of\u0000$B(ell^{p})/K(ell^{p})$ that preserves the Fredholm index. As a consequence,\u0000every separable $C^{*}$-algebra is isomorphic to a subalgebra of\u0000$B(ell^{p})/K(ell^{p})$. Another consequence is the existence of operators on\u0000$ell^{p}$ that behave like the essentially normal operators with arbitrary\u0000Fredholm indices in the Brown-Douglas-Fillmore theory.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195015","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}
Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming
Embezzlement of entanglement refers to the task of extracting entanglement from an entanglement resource via local operations and without communication while perturbing the resource arbitrarily little. Recently, the existence of embezzling states of bipartite systems of type III von Neumann algebras was shown. However, both the multipartite case and the precise relation between embezzling states and the notion of embezzling families, as originally defined by van Dam and Hayden, was left open. Here, we show that finite-dimensional approximations of multipartite embezzling states form multipartite embezzling families. In contrast, not every embezzling family converges to an embezzling state. We identify an additional consistency condition that ensures that an embezzling family converges to an embezzling state. This criterion distinguishes the embezzling family of van Dam and Hayden from the one by Leung, Toner, and Watrous. The latter generalizes to the multipartite setting. By taking a limit, we obtain a multipartite system of commuting type III$_1$ factors on which every state is an embezzling state. We discuss our results in the context of quantum field theory and quantum many-body physics. As open problems, we ask whether vacua of relativistic quantum fields in more than two spacetime dimensions are multipartite embezzling states and whether multipartite embezzlement allows for an operator-algebraic characterization.
纠缠的 "盗用"（Embezzlement of entanglement）是指通过局部操作从纠缠资源中提取纠缠，而不进行通信，同时对资源进行任意小的扰动。最近，有人证明了 III 型冯-诺依曼代数的双元系统存在 "侵吞 "状态。然而，在多方系统的情况下，embezzling 状态与 van Dam 和 Hayden 最初定义的 embezzling 族概念之间的确切关系却一直悬而未决。在这里，我们证明了多方贪污状态的有限维近似构成了多方贪污家族。相反，并非每个贪污家族都会收敛到贪污状态。我们确定了一个额外的一致性条件，以确保贪污家族趋同于贪污状态。这一标准将范达姆和海登的贪污家族与梁、托纳和沃特鲁斯的贪污家族区分开来。通过取一个极限，我们得到了一个多方III$_1$型共轭因子系统，在这个系统上，每个状态都是贪污状态。我们将在量子场论和量子多体物理学的背景下讨论我们的结果。作为开放性问题，我们提出在超过两个时空维度的相对论量子场虚空是否是多方侵吞态，以及多方侵吞态是否允许算子代数特性。
{"title":"Multipartite Embezzlement of Entanglement","authors":"Lauritz van Luijk, Alexander Stottmeister, Henrik Wilming","doi":"arxiv-2409.07646","DOIUrl":"https://doi.org/arxiv-2409.07646","url":null,"abstract":"Embezzlement of entanglement refers to the task of extracting entanglement\u0000from an entanglement resource via local operations and without communication\u0000while perturbing the resource arbitrarily little. Recently, the existence of\u0000embezzling states of bipartite systems of type III von Neumann algebras was\u0000shown. However, both the multipartite case and the precise relation between\u0000embezzling states and the notion of embezzling families, as originally defined\u0000by van Dam and Hayden, was left open. Here, we show that finite-dimensional\u0000approximations of multipartite embezzling states form multipartite embezzling\u0000families. In contrast, not every embezzling family converges to an embezzling\u0000state. We identify an additional consistency condition that ensures that an\u0000embezzling family converges to an embezzling state. This criterion\u0000distinguishes the embezzling family of van Dam and Hayden from the one by\u0000Leung, Toner, and Watrous. The latter generalizes to the multipartite setting.\u0000By taking a limit, we obtain a multipartite system of commuting type III$_1$\u0000factors on which every state is an embezzling state. We discuss our results in\u0000the context of quantum field theory and quantum many-body physics. As open\u0000problems, we ask whether vacua of relativistic quantum fields in more than two\u0000spacetime dimensions are multipartite embezzling states and whether\u0000multipartite embezzlement allows for an operator-algebraic characterization.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224975","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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