{"title":"KK-rigidity of simple nuclear C*-algebras","authors":"Christopher Schafhauser","doi":"arxiv-2408.02745","DOIUrl":"https://doi.org/arxiv-2408.02745","url":null,"abstract":"It is shown that if $A$ and $B$ are unital separable simple nuclear $mathcal\u0000Z$-stable C$^*$-algebras and there is a unital embedding $A rightarrow B$\u0000which is invertible on $KK$-theory and traces, then $A cong B$. In particular,\u0000two unital separable simple nuclear $mathcal Z$-stable C$^*$-algebras which\u0000either have real rank zero or unique trace are isomorphic if and only if they\u0000are homotopy equivalent. It is further shown that two finite strongly\u0000self-absorbing C$^*$-algebras are isomorphic if and only if they are\u0000$KK$-equivalent in a unit-preserving way.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930500","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":"Outer actions of finite groups on prime C*-algebras","authors":"Costel Peligrad","doi":"arxiv-2408.02510","DOIUrl":"https://doi.org/arxiv-2408.02510","url":null,"abstract":"An action of a compact, in particular finite group on a C*-algebra is called\u0000properly outer if no automorphism of the group that is distinct from identity\u0000is implemented by a unitary element of the algebra of local multipliers of the\u0000C*-algebra and strictly outer if the commutant of the algebra in the algebra of\u0000local mutipliers of the cross product consists of scalars [11]. In [11, Theorem\u000011] I proved that for finite groups and prime C*-algebras (not necessarily\u0000separable), the two notions are equivalent. I also proved that for finite\u0000abelian groups this is equivalent to other relevant properties of the action\u0000[11 Theorem 14]. In this paper I add other properties to the list in [11,\u0000Theorem 14].","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930510","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":"Capacities of quantum Markovian noise for large times","authors":"Omar Fawzi, Mizanur Rahaman, Mostafa Taheri","doi":"arxiv-2408.00116","DOIUrl":"https://doi.org/arxiv-2408.00116","url":null,"abstract":"Given a quantum Markovian noise model, we study the maximum dimension of a\u0000classical or quantum system that can be stored for arbitrarily large time. We\u0000show that, unlike the fixed time setting, in the limit of infinite time, the\u0000classical and quantum capacities are characterized by efficiently computable\u0000properties of the peripheral spectrum of the quantum channel. In addition, the\u0000capacities are additive under tensor product, which implies in the language of\u0000Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the\u0000same. We also provide an improved algorithm for computing the structure of the\u0000peripheral subspace of a quantum channel, which might be of independent\u0000interest.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882189","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":"Abstract semiclassical analysis of the van Hove model","authors":"Marco Falconi, Lorenzo Fratini","doi":"arxiv-2407.20603","DOIUrl":"https://doi.org/arxiv-2407.20603","url":null,"abstract":"In this paper we study the semiclassical limit $hslashto 0$ of a completely\u0000solvable model in quantum field theory: the van Hove model, describing a scalar\u0000field created and annihilated by an immovable source. Despite its simplicity,\u0000the van Hove model possesses many characterizing features of quantum fields,\u0000especially in the infrared region. In particular, the existence of non-Fock\u0000ground and equilibrium states in the presence of infrared singular sources\u0000makes a representation-independent algebraic approach of utmost importance. We\u0000make use of recent representation-independent techniques of infinite\u0000dimensional semiclassical analysis to establish the Bohr correspondence\u0000principle for the dynamics, equilibrium states, and long-time asymptotics in\u0000the van Hove model.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867287","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":"Pairs of fixed points for a class of operators on Hilbert spaces","authors":"A. Mokhtari, K. Saoudi, D. D. Repovš","doi":"arxiv-2407.17128","DOIUrl":"https://doi.org/arxiv-2407.17128","url":null,"abstract":"In this paper, existence of pairs of solutions is obtained for compact\u0000potential operators on Hilbert spaces. An application to a second-order\u0000boundary value problem is also given as an illustration of our results.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786256","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":"Invariant subspaces of perturbed backward shift","authors":"Soma Das, Jaydeb Sarkar","doi":"arxiv-2407.17352","DOIUrl":"https://doi.org/arxiv-2407.17352","url":null,"abstract":"We represent closed subspaces of the Hardy space that are invariant under\u0000finite-rank perturbations of the backward shift. We apply this to classify\u0000almost invariant subspaces of the backward shift and represent closed subspaces\u0000that are invariant under a more refined version of nearly invariant subspaces\u0000of the backward shift. Kernels of certain perturbed Toeplitz operators are\u0000examples of the newly introduced nearly invariant subspaces.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786344","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":"Maximal UHF subalgebras of certain C*-algebras","authors":"Nasser Golestani, Saeid Maleki Oche","doi":"arxiv-2407.17004","DOIUrl":"https://doi.org/arxiv-2407.17004","url":null,"abstract":"A well-known result in dynamical systems asserts that any Cantor minimal\u0000system $(X,T)$ has a maximal rational equicontinuous factor $(Y,S)$ which is in\u0000fact an odometer, and realizes the rational subgroup of the $K_0$-group of\u0000$(X,T)$, that is, $mathbb{Q}(K_0(X,T), 1) cong K^0(Y,S)$. We introduce the\u0000notion of a maximal UHF subalgebra and use it to obtain the C*-algebraic alonog\u0000of this result. We say a UHF subalgebra $B$ of a unital C*-algebra $A$ is a\u0000maximal UHF subalgebra if it contains the unit of $A$ any other such\u0000C*-subalgebra embeds unitaly into $B$. We prove that if $K_0(A)$ is\u0000unperforated and has a certain $K_0$-lifting property, then $B$ exists and is\u0000unique up to isomorphism, in particular, all simple separable unital\u0000C*-algebras with tracial rank zero and all unital Kirchberg algebras whose\u0000$K_0$-groups are unperforated, have a maximal UHF subalgebra. Not every unital\u0000C*-algebra has a maximal UHF subalgebra, for instance, the unital universal\u0000free product $mathrm{M}_2 ast_{r} mathrm{M}_3$. As an application, we give a\u0000C*-algebraic realization of the rational subgroup $mathbb{Q}(G,u)$ of any\u0000dimension group $G$ with order unit $u$, that is, there is a simple unital AF\u0000algebra (and a unital Kirchberg algebra) $A$ with a maximal UHF subalgebra $B$\u0000such that $(G,u)cong (K_0(A), [1]_0)$ and and $mathbb{Q}(G,u)cong K_0(B)$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786101","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":"Wiener pairs of Banach algebras of operator-valued matrices","authors":"Lukas Köhldorfer, Peter Balazs","doi":"arxiv-2407.16416","DOIUrl":"https://doi.org/arxiv-2407.16416","url":null,"abstract":"In this article we introduce several new examples of Wiener pairs\u0000$mathcal{A} subseteq mathcal{B}$, where $mathcal{B} =\u0000mathcal{B}(ell^2(X;mathcal{H}))$ is the Banach algebra of bounded operators\u0000acting on the Hilbert space-valued Bochner sequence space\u0000$ell^2(X;mathcal{H})$ and $mathcal{A} = mathcal{A}(X)$ is a Banach algebra\u0000consisting of operator-valued matrices indexed by some relatively separated set\u0000$X subset mathbb{R}^d$. In particular, we introduce\u0000$mathcal{B}(mathcal{H})$-valued versions of the Jaffard algebra, of certain\u0000weighted Schur-type algebras, of Banach algebras which are defined by more\u0000general off-diagonal decay conditions than polynomial decay, of weighted\u0000versions of the Baskakov-Gohberg-Sj\"ostrand algebra, and of anisotropic\u0000variations of all of these matrix algebras, and show that they are\u0000inverse-closed in $mathcal{B}(ell^2(X;mathcal{H}))$. In addition, we obtain\u0000that each of these Banach algebras is symmetric.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784057","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":"AF Embeddability of the C*-Algebra of a Deaconu-Renault Groupoid","authors":"Rafael Pereira Lima","doi":"arxiv-2407.16510","DOIUrl":"https://doi.org/arxiv-2407.16510","url":null,"abstract":"We study Deaconu-Renault groupoids corresponding to surjective local\u0000homeomorphisms on locally compact, Hausdorff, second countable, totally\u0000disconnected spaces, and we characterise when the C*-algebras of these\u0000groupoids are AF embeddable. Our main result generalises theorems in the\u0000literature for graphs and for crossed products of commutative C*-algebras by\u0000the integers. We give a condition on the surjective local homeomorphism that\u0000characterises the AF embeddability of the C*-algebra of the associated\u0000Deaconu-Renault groupoid. In order to prove our main result, we analyse\u0000homology groups for AF groupoids, and we prove a theorem that gives an explicit\u0000formula for the isomorphism of these groups and the corresponding K-theory.\u0000This isomorphism generalises Farsi, Kumjian, Pask, Sims (M\"unster J. Math,\u00002019) and Matui (Proc. Lond. Math. Soc, 2012), since we give an explicit\u0000formula for the isomorphism and we show that it preserves positive elements.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141786103","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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