Pub Date : 2022-06-15DOI: 10.7900/jot.2020dec04.2331
Kun Wang
The class of AH algebras with the ideal property and no dimension growth is classified by the invariant inv(⋅). In this paper, we introduce a new invariant, Inv(⋅), a refined version of inv(⋅) and show that they are equivalent for AH algebras with the ideal property and no dimension growth. Then we give a sufficient condition under which the Hausdorffified algebraic K1 group could be recovered from the traditional Elliott invariant. As an application, the class of AH algebras with the ideal property, no dimension growth, and finitely many ideals can be classified by the extended Elliott invariant.
{"title":"Classification of AH algebras with finitely many ideals","authors":"Kun Wang","doi":"10.7900/jot.2020dec04.2331","DOIUrl":"https://doi.org/10.7900/jot.2020dec04.2331","url":null,"abstract":"The class of AH algebras with the ideal property and no dimension growth is classified by the invariant inv(⋅). In this paper, we introduce a new invariant, Inv(⋅), a refined version of inv(⋅) and show that they are equivalent for AH algebras with the ideal property and no dimension growth. Then we give a sufficient condition under which the Hausdorffified algebraic K1 group could be recovered from the traditional Elliott invariant. As an application, the class of AH algebras with the ideal property, no dimension growth, and finitely many ideals can be classified by the extended Elliott invariant.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43634358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-15DOI: 10.7900/jot.2020dec21.2321
Samuel Evington
We obtain an improved upper bound for the nuclear dimension of extensions of O∞-stable C∗-algebras. In particular, we prove that the nuclear dimension of a full extension of an O∞-stable C∗-algebra by a stable AF algebra is one.
{"title":"Nuclear dimension of extensions of O∞-stable algebras","authors":"Samuel Evington","doi":"10.7900/jot.2020dec21.2321","DOIUrl":"https://doi.org/10.7900/jot.2020dec21.2321","url":null,"abstract":"We obtain an improved upper bound for the nuclear dimension of extensions of O∞-stable C∗-algebras. In particular, we prove that the nuclear dimension of a full extension of an O∞-stable C∗-algebra by a stable AF algebra is one.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47441665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-15DOI: 10.7900/jot.2020nov28.2310
Xinqi Guo, Maofa Wang
In this paper, we study the compactness of any finite linear combination of composition operators with general symbols on weighted Bergman spaces over the unit ball in terms of a power type criterion. The strategy of the proof involves the subtle connection of composition operator theory between weighted Bergman spaces and Korenblum spaces.
{"title":"Compact linear combinations of composition operators over the unit ball","authors":"Xinqi Guo, Maofa Wang","doi":"10.7900/jot.2020nov28.2310","DOIUrl":"https://doi.org/10.7900/jot.2020nov28.2310","url":null,"abstract":"In this paper, we study the compactness of any finite linear combination of composition operators with general symbols on weighted Bergman spaces over the unit ball in terms of a power type criterion. The strategy of the proof involves the subtle connection of composition operator theory between weighted Bergman spaces and Korenblum spaces.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49021835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-15DOI: 10.7900/jot.2020nov17.2322
A. Pyshkin
In 1990s several papers studied a strong tridiagonal M-basis that did not possess rank one density property. We offer a new method for the study of more generic finite-band M-bases, employing a graph theory framework. We determine the necessary and sufficient conditions for rank one density property in this class of M-bases. Also we give some sufficient conditions concerning k point density property.
{"title":"Rank one density for a class of M-bases","authors":"A. Pyshkin","doi":"10.7900/jot.2020nov17.2322","DOIUrl":"https://doi.org/10.7900/jot.2020nov17.2322","url":null,"abstract":"In 1990s several papers studied a strong tridiagonal M-basis that did not possess rank one density property. We offer a new method for the study of more generic finite-band M-bases, employing a graph theory framework. We determine the necessary and sufficient conditions for rank one density property in this class of M-bases. Also we give some sufficient conditions concerning k point density property.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41766420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-15DOI: 10.7900/jot.2020oct20.2297
J. Bassi, F. Rădulescu
We exhibit examples of separable boundaries for nonhyperbolic groups. The main ingredient is the alignment property introduced by Furman in the study of rigidity properties of discrete subgroups of algebraic groups.
{"title":"Separable boundaries for nonhyperbolic groups","authors":"J. Bassi, F. Rădulescu","doi":"10.7900/jot.2020oct20.2297","DOIUrl":"https://doi.org/10.7900/jot.2020oct20.2297","url":null,"abstract":"We exhibit examples of separable boundaries for nonhyperbolic groups. The main ingredient is the alignment property introduced by Furman in the study of rigidity properties of discrete subgroups of algebraic groups.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41519769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-15DOI: 10.7900/jot.2020sep30.2352
Wei Shi, Junhao Shen, Weichen Gu, Minghui Ma
Let M be a factor of type II with a faithful normal semifinite tracial weight τ, and P the set of all projections in M. Denote by Pc the Grassmann space of all projections in P with trace c, where c is a positive real number. The aim of this paper is to describe the general form of Lp-isometries between Grassmann spaces in a factor of type II. Moreover, we prove that, when $0
{"title":"Lp-isometries of Grassmann spaces in factors of type II","authors":"Wei Shi, Junhao Shen, Weichen Gu, Minghui Ma","doi":"10.7900/jot.2020sep30.2352","DOIUrl":"https://doi.org/10.7900/jot.2020sep30.2352","url":null,"abstract":"Let M be a factor of type II with a faithful normal semifinite tracial weight τ, and P the set of all projections in M. Denote by Pc the Grassmann space of all projections in P with trace c, where c is a positive real number. The aim of this paper is to describe the general form of Lp-isometries between Grassmann spaces in a factor of type II. Moreover, we prove that, when $0","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42120498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-15DOI: 10.7900/jot.2020jun29.2289
Gelu Popescu
Given a set Q of polynomials in noncommutative indeterminates Z1,…,Zn and a regular domain Dmp(H)⊂B(H)n, m,n∈N, associated with a positive regular polynomial p∈C⟨Z1,…,Zn⟩, we consider the variety VQ(H):={X=(X1,…,Xn)∈Dmp(H):g(X)=0 for all g∈Q}. Each variety VQ(H) admits a {it universal model} B=(B1,…,Bn). The main goal of the paper is to study the structure of the ∗-representations of the C∗-algebra C∗(VQ) generated by B1,…,Bn and the identity.
{"title":"Wold decompositions for representations of C∗-algebras associated with noncommutative varieties","authors":"Gelu Popescu","doi":"10.7900/jot.2020jun29.2289","DOIUrl":"https://doi.org/10.7900/jot.2020jun29.2289","url":null,"abstract":"Given a set Q of polynomials in noncommutative indeterminates Z1,…,Zn and a regular domain Dmp(H)⊂B(H)n, m,n∈N, associated with a positive regular polynomial p∈C⟨Z1,…,Zn⟩, we consider the variety VQ(H):={X=(X1,…,Xn)∈Dmp(H):g(X)=0 for all g∈Q}. Each variety VQ(H) admits a {it universal model} B=(B1,…,Bn). The main goal of the paper is to study the structure of the ∗-representations of the C∗-algebra C∗(VQ) generated by B1,…,Bn and the identity.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48482547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-15DOI: 10.7900/jot.202jul23.2345
Konstantinos A. Beros, P. Larson
Each w∈ℓ∞ defines a bacwards weighted shift Bw:ℓ2→ℓ2. A vector x∈ℓ2 is textit{hypercyclic} for Bw if the set of forward iterates of x is dense in ℓ2. For each such w, the set HC(w) consisting of all vectors hypercyclic for Bw is Gδ. The set of textit{common hypercyclic vectors} for a set W⊆ℓ∞ is the set HC∗(W)=⋂w∈WHC(w). We show that HC∗(W) can be made arbitrarily complicated by making W sufficiently complex, and that even for a Gδ set W the set HC∗(W) can be non-Borel. Finally, by assuming the continuum hypothesis or Martin's axiom, we are able to construct a set W such that HC∗(W) does not have the property of Baire.
每个w∈r∞定义了一个向后加权移位Bw: r 2→r 2。如果x的前向迭代集在l2中是密集的，则向量x∈l2对于Bw是textit{超循环}的。对于每一个这样的w，由Bw的所有超环向量组成的集合HC(w)为Gδ。集W∈WHC(W)的textit{公共超循环向量}集为集HC∗(W)= W∈WHC(W)。我们证明了HC∗(W)可以通过使W足够复杂而变得任意复杂，并且即使对于Gδ集W，集HC∗(W)也可以是非borel的。最后，通过假设连续统假设或马丁公理，我们能够构造一个集合W，使得HC * (W)不具有贝尔性质。
{"title":"Common hypercyclic vectors for unilateral weighted shifts on ℓ2","authors":"Konstantinos A. Beros, P. Larson","doi":"10.7900/jot.202jul23.2345","DOIUrl":"https://doi.org/10.7900/jot.202jul23.2345","url":null,"abstract":"Each w∈ℓ∞ defines a bacwards weighted shift Bw:ℓ2→ℓ2. A vector x∈ℓ2 is textit{hypercyclic} for Bw if the set of forward iterates of x is dense in ℓ2. For each such w, the set HC(w) consisting of all vectors hypercyclic for Bw is Gδ. The set of textit{common hypercyclic vectors} for a set W⊆ℓ∞ is the set HC∗(W)=⋂w∈WHC(w). We show that HC∗(W) can be made arbitrarily complicated by making W sufficiently complex, and that even for a Gδ set W the set HC∗(W) can be non-Borel. Finally, by assuming the continuum hypothesis or Martin's axiom, we are able to construct a set W such that HC∗(W) does not have the property of Baire.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41597945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-15DOI: 10.7900/jot.2020apr30.2294
W. Arendt, I. Chalendar, B. Moletsane
In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy space generalizing results from Semigroup Forum 95(2017), 281-292. A holomorphic semigroup operating on the Hardy space is obtained whose asymptotic behaviour is studied and which is compared with the semigroup generated by the elliptic operator with periodic boundary conditions on L2(0,2π).
{"title":"Perturbation by weakly continuous forms and semigroups on Hardy space","authors":"W. Arendt, I. Chalendar, B. Moletsane","doi":"10.7900/jot.2020apr30.2294","DOIUrl":"https://doi.org/10.7900/jot.2020apr30.2294","url":null,"abstract":"In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy space generalizing results from Semigroup Forum 95(2017), 281-292. A holomorphic semigroup operating on the Hardy space is obtained whose asymptotic behaviour is studied and which is compared with the semigroup generated by the elliptic operator with periodic boundary conditions on L2(0,2π).","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47333942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-15DOI: 10.7900/jot.2020apr21.2306
G. Elliott, Zhichao Liu
Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.
{"title":"Distance between unitary orbits in C∗-algebras with stable rank one and real rank zero","authors":"G. Elliott, Zhichao Liu","doi":"10.7900/jot.2020apr21.2306","DOIUrl":"https://doi.org/10.7900/jot.2020apr21.2306","url":null,"abstract":"Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42930290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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