Pub Date : 2020-04-06DOI: 10.7900/jot.2020jan20.2267
M. Asadi, M. A. Asadi-Vasfi
Let X be a compact metric space, let A be a unital AH-algebra with large matrix sizes, and let B be a stably finite unital C∗-algebra. Then we give a lower bound for the radius of comparison of C(X)⊗B and prove that the dimension-rank ratio satisfies drr(A)=drr(C(X)⊗A). We also give a class of unital AH-algebras A with rc(C(X)⊗A)=rc(A). We further give a class of stably finite exact Z-stable unital C∗-algebras with nonzero radius of comparison.
{"title":"The radius of comparison of the tensor product of a C∗-algebra with C(X)","authors":"M. Asadi, M. A. Asadi-Vasfi","doi":"10.7900/jot.2020jan20.2267","DOIUrl":"https://doi.org/10.7900/jot.2020jan20.2267","url":null,"abstract":"Let X be a compact metric space, let A be a unital AH-algebra with large matrix sizes, and let B be a stably finite unital C∗-algebra. Then we give a lower bound for the radius of comparison of C(X)⊗B and prove that the dimension-rank ratio satisfies drr(A)=drr(C(X)⊗A). We also give a class of unital AH-algebras A with rc(C(X)⊗A)=rc(A). We further give a class of stably finite exact Z-stable unital C∗-algebras with nonzero radius of comparison.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48793862","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 : 2020-04-03DOI: 10.7900/jot.2020apr17.2281
R. Eskandari, M. Frank, V. Manuilov, M. Moslehian
We introduce the B-spline interpolation problem corresponding to a C∗-valued sesquilinear form on a Hilbert C∗-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert C∗-module is self-dual. Passing to the setting of Hilbert W∗-modules, we present our main result by characterizing when the spline interpolation problem for the extended C∗-valued sesquilinear form has a solution. Finally, solutions of the B-spline interpolation problem for Hilbert C∗-modules over C∗-ideals of W∗-algebras are extensively discussed.
引入Hilbert C * -模上C *值半线性形式对应的b样条插值问题，研究其基本性质及其解的唯一性。我们首先研究了Hilbert C *模是自对偶的情况下的问题。传递到Hilbert W * -模的集合，我们通过刻画扩展C *值半线性形式的样条插值问题何时有解来给出我们的主要结果。最后，广泛讨论了W *代数的C *理想上Hilbert C * -模的b样条插值问题的解。
{"title":"B-spline interpolation problem in Hilbert C∗-modules","authors":"R. Eskandari, M. Frank, V. Manuilov, M. Moslehian","doi":"10.7900/jot.2020apr17.2281","DOIUrl":"https://doi.org/10.7900/jot.2020apr17.2281","url":null,"abstract":"We introduce the B-spline interpolation problem corresponding to a C∗-valued sesquilinear form on a Hilbert C∗-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert C∗-module is self-dual. Passing to the setting of Hilbert W∗-modules, we present our main result by characterizing when the spline interpolation problem for the extended C∗-valued sesquilinear form has a solution. Finally, solutions of the B-spline interpolation problem for Hilbert C∗-modules over C∗-ideals of W∗-algebras are extensively discussed.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47339491","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 : 2020-04-01DOI: 10.7900/jot.2020may15.2286
S. Bose, P. Muthukumar, J. Sarkar
The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: characterize φ, holomorphic self maps of D, and inner functions θ∈H∞(D) such that the Beurling type invariant subspace θH2 is an invariant subspace for Cφ. We prove the following result: Cφ(θH2)⊆θH2 if and only if θ∘φθ∈S(D). This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.
{"title":"Beurling type invariant subspaces of composition operators","authors":"S. Bose, P. Muthukumar, J. Sarkar","doi":"10.7900/jot.2020may15.2286","DOIUrl":"https://doi.org/10.7900/jot.2020may15.2286","url":null,"abstract":"The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: characterize φ, holomorphic self maps of D, and inner functions θ∈H∞(D) such that the Beurling type invariant subspace θH2 is an invariant subspace for Cφ. We prove the following result: Cφ(θH2)⊆θH2 if and only if θ∘φθ∈S(D). This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44433384","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 : 2020-04-01DOI: 10.7900/jot.2020feb17.2269
E. Albrecht, B. Chevreau
We consider compact perturbations S=DΛ+K of normal diagonal operators DΛ by certain compact operators. Sufficient conditions for K to ensure the existence of non-trivial hyperinvariant subspaces for S have recently been obtained by Foiac{s} et al. in C. Foiac{s}, I.B. Jung, E. Ko, C. Pearcy, textit{J. Funct. Anal.} textbf{253}(2007), 628--646, C. Foiac{s}, I.B. Jung, E. Ko, C.~Pearcy, textit{Indiana Univ. Math. J.} textbf{57}(2008), 2745--2760, {C. Foiac{s}, I.B. Jung, E. Ko, C.Pearcy}, textit{J. Math. Anal. Appl.} textbf{375}(2011), 602--609 (followed by Fang--Xia textit{J. Funct. Anal} textbf{263}(2012), 135-1377, and Klaja textit{J. Operator Theory} textbf{73}(2015), 127--142, by using certain spectral integrals along straight lines through the spectrum of S. In this note, the authors use circular cuts and get positive results under less restrictive local conditions for K.
我们用某些紧算子考虑正常对角算子DΛ的紧摄动S=DΛ+K。最近，Foia c{s}等人在C. Foia c{s}, I.B. Jung, E. Ko, C. Pearcy, textit{J. Funct中得到了K保证S的非平凡超不变子空间存在的充分条件}。《数学》textbf{253}(2007)，628—646,C. Foia c{s}, I.B. Jung, E. Ko, C. Pearcy，textit{印第安纳大学数学。J.}textbf{57}(2008)， 2745—2760,{C. Foiac{s}, I.B. Jung, E. Ko, C. pearcy, }textit{J. Math。分析的[j]}textbf{.}中国科学:自然科学，2011(5)，349 - textit{349。Anal}textbf{263}(2012)， 135-1377，和Klaja textit{J.算子理论}textbf{73}(2015)，127- 142，通过使用s的谱沿直线的某些谱积分。在这篇笔记中，作者使用圆形切割并在较少限制的局部条件下得到K的正结果。
{"title":"Compact perturbations of scalar type spectral operators","authors":"E. Albrecht, B. Chevreau","doi":"10.7900/jot.2020feb17.2269","DOIUrl":"https://doi.org/10.7900/jot.2020feb17.2269","url":null,"abstract":"We consider compact perturbations S=DΛ+K of normal diagonal operators DΛ by certain compact operators. Sufficient conditions for K to ensure the existence of non-trivial hyperinvariant subspaces for S have recently been obtained by Foiac{s} et al. in C. Foiac{s}, I.B. Jung, E. Ko, C. Pearcy, textit{J. Funct. Anal.} textbf{253}(2007), 628--646, C. Foiac{s}, I.B. Jung, E. Ko, C.~Pearcy, textit{Indiana Univ. Math. J.} textbf{57}(2008), 2745--2760, {C. Foiac{s}, I.B. Jung, E. Ko, C.Pearcy}, textit{J. Math. Anal. Appl.} textbf{375}(2011), 602--609 (followed by Fang--Xia textit{J. Funct. Anal} textbf{263}(2012), 135-1377, and Klaja textit{J. Operator Theory} textbf{73}(2015), 127--142, by using certain spectral integrals along straight lines through the spectrum of S. In this note, the authors use circular cuts and get positive results under less restrictive local conditions for K.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43128931","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 : 2020-03-22DOI: 10.7900/jot.2020aug04.2328
Ilan Hirshberg
We investigate connections between actions on separable C∗-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra Z. We show that if A admits an approximately inner group action with finite Rokhlin dimension with commuting towers then A is Z-stable. We obtain analogous results for tracial version of the Rokhlin property and approximate innerness. Going beyond approximate innerness, for actions of a single automorphism which have the Rokhlin property and are almost periodic in a suitable sense, the crossed product absorbs Z even when the original algebra does not.
{"title":"Rokhlin-type properties, approximate innerness and Z-stability","authors":"Ilan Hirshberg","doi":"10.7900/jot.2020aug04.2328","DOIUrl":"https://doi.org/10.7900/jot.2020aug04.2328","url":null,"abstract":"We investigate connections between actions on separable C∗-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra Z. We show that if A admits an approximately inner group action with finite Rokhlin dimension with commuting towers then A is Z-stable. We obtain analogous results for tracial version of the Rokhlin property and approximate innerness. Going beyond approximate innerness, for actions of a single automorphism which have the Rokhlin property and are almost periodic in a suitable sense, the crossed product absorbs Z even when the original algebra does not.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46392496","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 : 2020-03-15DOI: 10.7900/jot.2018oct09.2224
Yi Wang, Jingbo Xia
Let Ω be a bounded, strongly pseudo-convex domain with smooth boundary in Cn. Suppose that h is an analytic function defined on an open set containing ¯¯¯¯Ω. We show that the principal submodule of the Hardy module H2(Ω) generated by h is p-essentially normal for p>n.
{"title":"Essential normality of principal submodules of the Hardy module on a strongly pseudo-convex domain","authors":"Yi Wang, Jingbo Xia","doi":"10.7900/jot.2018oct09.2224","DOIUrl":"https://doi.org/10.7900/jot.2018oct09.2224","url":null,"abstract":"Let Ω be a bounded, strongly pseudo-convex domain with smooth boundary in Cn. Suppose that h is an analytic function defined on an open set containing ¯¯¯¯Ω. We show that the principal submodule of the Hardy module H2(Ω) generated by h is p-essentially normal for p>n.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44814502","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 : 2020-03-15DOI: 10.7900/jot.2018nov07.2234
L. Marcoux
It is still an open question to know whether or not every quasidiagonal operator can be expressed as a norm-limit of algebraic quasidiagonal operators. In this note, we provide an alternative characterization of those operators which may be expressed as such limits, in the hope that this may lead to a solution of this problem.
{"title":"On norm-limits of algebraic quasidiagonal operators","authors":"L. Marcoux","doi":"10.7900/jot.2018nov07.2234","DOIUrl":"https://doi.org/10.7900/jot.2018nov07.2234","url":null,"abstract":"It is still an open question to know whether or not every quasidiagonal operator can be expressed as a norm-limit of algebraic quasidiagonal operators. In this note, we provide an alternative characterization of those operators which may be expressed as such limits, in the hope that this may lead to a solution of this problem.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41610018","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 : 2020-03-15DOI: 10.7900/jot.2018aug09.2252
A. Bourhim, J. Mashreghi
We show that a surjective map φ between two unital C∗-algebras A and B, with φ(0)=0, satisfies Λε(φ(x1)−φ(x2))=Λε(x1−x2),(x1, x2∈A), where Λε denotes the ε-pseudospectrum, if and only if φ is a Jordan ∗-isomor-phism. We also characterize maps φ1 and φ2 from A onto B that satisfy Λε(φ1(x1)φ2(x2))=Λε(x1x2),(x1, x2∈A), or some other binary operations, in terms of Jordan ∗-isomorphisms. The main results imply several other characterizations of Jordan ∗-isomorphisms which are interesting in their own right.
{"title":"Jordan maps and pseudospectrum in C∗-algebras","authors":"A. Bourhim, J. Mashreghi","doi":"10.7900/jot.2018aug09.2252","DOIUrl":"https://doi.org/10.7900/jot.2018aug09.2252","url":null,"abstract":"We show that a surjective map φ between two unital C∗-algebras A and B, with φ(0)=0, satisfies Λε(φ(x1)−φ(x2))=Λε(x1−x2),(x1, x2∈A), where Λε denotes the ε-pseudospectrum, if and only if φ is a Jordan ∗-isomor-phism. We also characterize maps φ1 and φ2 from A onto B that satisfy Λε(φ1(x1)φ2(x2))=Λε(x1x2),(x1, x2∈A), or some other binary operations, in terms of Jordan ∗-isomorphisms. The main results imply several other characterizations of Jordan ∗-isomorphisms which are interesting in their own right.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47148850","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 : 2020-02-22DOI: 10.7900/jot.2020sep27.2329
Bryan Goldberg, Rongwei Yang
This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.
{"title":"Self-similarity and spectral dynamics","authors":"Bryan Goldberg, Rongwei Yang","doi":"10.7900/jot.2020sep27.2329","DOIUrl":"https://doi.org/10.7900/jot.2020sep27.2329","url":null,"abstract":"This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46971442","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 : 2020-02-06DOI: 10.7900/jot.2020feb06.2268
R. Hagger
We study the Toeplitz algebra which is generated by Toeplitz operators with bounded symbols on the Fock space Fpα. We show that the Toeplitz algebra coincides with each of the algebras generated by band-dominated, sufficiently localized and weakly localized operators, respectively. Moreover, we determine its essential commutant and its essential bicommutant. For p=2 these results were obtained recently by Xia. However, Xia's ideas are mostly connected to Hilbert space theory and methods which are not applicable for p≠2. Instead, we use a recent result of Fulsche to generalize Xia's theorems.
{"title":"Essential commutants and characterizations of the Toeplitz algebra","authors":"R. Hagger","doi":"10.7900/jot.2020feb06.2268","DOIUrl":"https://doi.org/10.7900/jot.2020feb06.2268","url":null,"abstract":"We study the Toeplitz algebra which is generated by Toeplitz operators with bounded symbols on the Fock space Fpα. We show that the Toeplitz algebra coincides with each of the algebras generated by band-dominated, sufficiently localized and weakly localized operators, respectively. Moreover, we determine its essential commutant and its essential bicommutant. For p=2 these results were obtained recently by Xia. However, Xia's ideas are mostly connected to Hilbert space theory and methods which are not applicable for p≠2. Instead, we use a recent result of Fulsche to generalize Xia's theorems.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48530695","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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