It is shown that if $A$ and $B$ are unital separable simple nuclear $mathcal Z$-stable C$^*$-algebras and there is a unital embedding $A rightarrow B$ which is invertible on $KK$-theory and traces, then $A cong B$. In particular, two unital separable simple nuclear $mathcal Z$-stable C$^*$-algebras which either have real rank zero or unique trace are isomorphic if and only if they are homotopy equivalent. It is further shown that two finite strongly self-absorbing C$^*$-algebras are isomorphic if and only if they are $KK$-equivalent in a unit-preserving way.
{"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":null,"pages":null},"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}
An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra and strictly outer if the commutant of the algebra in the algebra of local mutipliers of the cross product consists of scalars [11]. In [11, Theorem 11] I proved that for finite groups and prime C*-algebras (not necessarily separable), the two notions are equivalent. I also proved that for finite abelian groups this is equivalent to other relevant properties of the action [11 Theorem 14]. In this paper I add other properties to the list in [11, Theorem 14].
{"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":null,"pages":null},"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}
A Banach space characterization of simple real or complex $C^*$-algebras is given which even characterizes the underlying field. As an application, it is shown that if $mathfrak A_1$ and $mathfrak A_2$ are Birkhoff-James isomorphic simple $C^*$-algebras over the fields $mathbb F_1$ and $mathbb F_2$, respectively and if $mathfrak A_1$ is finite-dimensional with dimension greater than one, then $mathbb F_1=mathbb F_2$ and $mathfrak A_1$ and $mathfrak A_2$ are (isometrically) $ast$-isomorphic $C^*$-algebras.
{"title":"Non-linear classification of finite-dimensional simple $C^*$-algebras","authors":"Bojan Kuzma, Sushil Singla","doi":"arxiv-2407.21582","DOIUrl":"https://doi.org/arxiv-2407.21582","url":null,"abstract":"A Banach space characterization of simple real or complex $C^*$-algebras is\u0000given which even characterizes the underlying field. As an application, it is\u0000shown that if $mathfrak A_1$ and $mathfrak A_2$ are Birkhoff-James isomorphic\u0000simple $C^*$-algebras over the fields $mathbb F_1$ and $mathbb F_2$,\u0000respectively and if $mathfrak A_1$ is finite-dimensional with dimension\u0000greater than one, then $mathbb F_1=mathbb F_2$ and $mathfrak A_1$ and\u0000$mathfrak A_2$ are (isometrically) $ast$-isomorphic $C^*$-algebras.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867222","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}
Given a quantum Markovian noise model, we study the maximum dimension of a classical or quantum system that can be stored for arbitrarily large time. We show that, unlike the fixed time setting, in the limit of infinite time, the classical and quantum capacities are characterized by efficiently computable properties of the peripheral spectrum of the quantum channel. In addition, the capacities are additive under tensor product, which implies in the language of Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the same. We also provide an improved algorithm for computing the structure of the peripheral subspace of a quantum channel, which might be of independent interest.
{"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":null,"pages":null},"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}
In this paper we study the semiclassical limit $hslashto 0$ of a completely solvable model in quantum field theory: the van Hove model, describing a scalar field created and annihilated by an immovable source. Despite its simplicity, the van Hove model possesses many characterizing features of quantum fields, especially in the infrared region. In particular, the existence of non-Fock ground and equilibrium states in the presence of infrared singular sources makes a representation-independent algebraic approach of utmost importance. We make use of recent representation-independent techniques of infinite dimensional semiclassical analysis to establish the Bohr correspondence principle for the dynamics, equilibrium states, and long-time asymptotics in the van Hove model.
{"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":null,"pages":null},"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}
We compute the automorphism group of the intersection graph of many large-type Artin groups. This graph is an analogue of the curve graph of mapping class groups but in the context of Artin groups. As an application, we deduce a number of rigidity and classification results for these groups, including computation of outer automorphism groups, commensurability classification, quasi-isometric rigidity, measure equivalence rigidity, orbit equivalence rigidity, rigidity of lattice embedding, and rigidity of cross-product von Neumann algebra.
{"title":"Rigidity and classification results for large-type Artin groups","authors":"Jingyin Huang, Damian Osajda, Nicolas Vaskou","doi":"arxiv-2407.19940","DOIUrl":"https://doi.org/arxiv-2407.19940","url":null,"abstract":"We compute the automorphism group of the intersection graph of many\u0000large-type Artin groups. This graph is an analogue of the curve graph of\u0000mapping class groups but in the context of Artin groups. As an application, we\u0000deduce a number of rigidity and classification results for these groups,\u0000including computation of outer automorphism groups, commensurability\u0000classification, quasi-isometric rigidity, measure equivalence rigidity, orbit\u0000equivalence rigidity, rigidity of lattice embedding, and rigidity of\u0000cross-product von Neumann algebra.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141867288","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, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results.
{"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":null,"pages":null},"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}
We represent closed subspaces of the Hardy space that are invariant under finite-rank perturbations of the backward shift. We apply this to classify almost invariant subspaces of the backward shift and represent closed subspaces that are invariant under a more refined version of nearly invariant subspaces of the backward shift. Kernels of certain perturbed Toeplitz operators are examples of the newly introduced nearly invariant subspaces.
{"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":null,"pages":null},"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}
A well-known result in dynamical systems asserts that any Cantor minimal system $(X,T)$ has a maximal rational equicontinuous factor $(Y,S)$ which is in fact an odometer, and realizes the rational subgroup of the $K_0$-group of $(X,T)$, that is, $mathbb{Q}(K_0(X,T), 1) cong K^0(Y,S)$. We introduce the notion of a maximal UHF subalgebra and use it to obtain the C*-algebraic alonog of this result. We say a UHF subalgebra $B$ of a unital C*-algebra $A$ is a maximal UHF subalgebra if it contains the unit of $A$ any other such C*-subalgebra embeds unitaly into $B$. We prove that if $K_0(A)$ is unperforated and has a certain $K_0$-lifting property, then $B$ exists and is unique up to isomorphism, in particular, all simple separable unital C*-algebras with tracial rank zero and all unital Kirchberg algebras whose $K_0$-groups are unperforated, have a maximal UHF subalgebra. Not every unital C*-algebra has a maximal UHF subalgebra, for instance, the unital universal free product $mathrm{M}_2 ast_{r} mathrm{M}_3$. As an application, we give a C*-algebraic realization of the rational subgroup $mathbb{Q}(G,u)$ of any dimension group $G$ with order unit $u$, that is, there is a simple unital AF algebra (and a unital Kirchberg algebra) $A$ with a maximal UHF subalgebra $B$ such that $(G,u)cong (K_0(A), [1]_0)$ and and $mathbb{Q}(G,u)cong K_0(B)$.
{"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":null,"pages":null},"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}
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