{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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
{"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":null,"pages":null},"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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