Pub Date : 2020-07-02DOI: 10.7900/jot.2020sep17.2307
A. Chirvasitu
We prove a number of results on the automorphisms and isomorphisms between Hardy--Toeplitz algebras T(D) associated to bounded symmetric domains D: that the stable isomorphism class of T(D) determines D (even when it is reducible), that for reducible domains D=D1×⋯×Ds the automorphisms of the Shilov boundary ˇS(D) induced by those of T(D) permute the Shilov boundaries ˇS(Di), and that by contrast to arbitrary solvable algebras, automorphisms of T(D) that are trivial on their character spaces ˇS(D) are trivial on the entire spectrum ˆT(D).
{"title":"Rigidity results for automorphisms of Hardy--Toeplitz C∗-algebras","authors":"A. Chirvasitu","doi":"10.7900/jot.2020sep17.2307","DOIUrl":"https://doi.org/10.7900/jot.2020sep17.2307","url":null,"abstract":"We prove a number of results on the automorphisms and isomorphisms between Hardy--Toeplitz algebras T(D) associated to bounded symmetric domains D: that the stable isomorphism class of T(D) determines D (even when it is reducible), that for reducible domains D=D1×⋯×Ds the automorphisms of the Shilov boundary ˇS(D) induced by those of T(D) permute the Shilov boundaries ˇS(Di), and that by contrast to arbitrary solvable algebras, automorphisms of T(D) that are trivial on their character spaces ˇS(D) are trivial on the entire spectrum ˆT(D).","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46691247","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 : 2020-06-18DOI: 10.7900/jot.2020jul02.2313
L. Helmer, B. Solel
We study the C∗-algebra T/K where T is the C∗-algebra generated by d weighted shifts on the Fock space of Cd, F(Cd), (where the weights are given by a sequence {Zk} of matrices Zk∈Mdk(C)) and K is the algebra of compact operators on the Fock space. If Zk=I for every k, T/K is the Cuntz algebra Od. We show that T/K is isomorphic to a Cuntz--Pimsner algebra and use it to find conditions for the algebra to be simple. We present examples of simple and of nonsimple algebras of this type. We also describe the C∗-representations of T/K.
{"title":"Weighted Cuntz algebras","authors":"L. Helmer, B. Solel","doi":"10.7900/jot.2020jul02.2313","DOIUrl":"https://doi.org/10.7900/jot.2020jul02.2313","url":null,"abstract":"We study the C∗-algebra T/K where T is the C∗-algebra generated by d weighted shifts on the Fock space of Cd, F(Cd), (where the weights are given by a sequence {Zk} of matrices Zk∈Mdk(C)) and K is the algebra of compact operators on the Fock space. If Zk=I for every k, T/K is the Cuntz algebra Od. We show that T/K is isomorphic to a Cuntz--Pimsner algebra and use it to find conditions for the algebra to be simple. We present examples of simple and of nonsimple algebras of this type. We also describe the C∗-representations of T/K.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46544927","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 : 2020-06-15DOI: 10.7900/jot.2020aug30.2315
P. Jolissaint
Let M⊂B(H) be a von Neumann algebra acting on the (separable) Hilbert space H. We first prove that M is finite if and only if, for every x∈M and for all vectors ξ,η∈H, the coefficient function u↦⟨uxu∗ξ|η⟩ is weakly almost periodic on the topological group UM of unitaries in M (equipped with the weak operator topology). The main device is the unique invariant mean on the C∗-algebra WAP(UM) of weakly almost periodic functions on UM. Next, we prove that every coefficient function u↦⟨uxu∗ξ|η⟩ is almost periodic if and only if M is a direct sum of a diffuse, abelian von Neumann algebra and finite-dimensional factors. Incidentally, we prove that if M is a diffuse von Neumann algebra, then its unitary group is minimally almost periodic.
{"title":"Almost and weakly almost periodic functions on the unitary groups of von Neumann algebras","authors":"P. Jolissaint","doi":"10.7900/jot.2020aug30.2315","DOIUrl":"https://doi.org/10.7900/jot.2020aug30.2315","url":null,"abstract":"Let M⊂B(H) be a von Neumann algebra acting on the (separable) Hilbert space H. We first prove that M is finite if and only if, for every x∈M and for all vectors ξ,η∈H, the coefficient function u↦⟨uxu∗ξ|η⟩ is weakly almost periodic on the topological group UM of unitaries in M (equipped with the weak operator topology). The main device is the unique invariant mean on the C∗-algebra WAP(UM) of weakly almost periodic functions on UM. Next, we prove that every coefficient function u↦⟨uxu∗ξ|η⟩ is almost periodic if and only if M is a direct sum of a diffuse, abelian von Neumann algebra and finite-dimensional factors. Incidentally, we prove that if M is a diffuse von Neumann algebra, then its unitary group is minimally almost periodic.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48637661","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 : 2020-06-04DOI: 10.7900/jot.2020sep23.2312
Jacek Krajczok
The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on G and ˆG act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group ˆSUq(2) and the quantum az+b group.
{"title":"Modular properties of type I locally compact quantum groups","authors":"Jacek Krajczok","doi":"10.7900/jot.2020sep23.2312","DOIUrl":"https://doi.org/10.7900/jot.2020sep23.2312","url":null,"abstract":"The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on G and ˆG act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group ˆSUq(2) and the quantum az+b group.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45182322","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 : 2020-05-15DOI: 10.7900/jot.2019may21.2264
O. Szehr, R. Zarouf
We introduce a ``dual-space approach'' to mixed Nevanlinna--Pick Carath'eodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting one of D. Sarason and B. Nagy-C. Foiac{s}. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This functional extension lemma plays a similar role as Sarason's commutant lifting theorem but it only involves the predual of X and no Hilbert space structure is needed. As an example, we present the respective Pick-type interpolation theorems for Beurling-Sobolev spaces.
{"title":"Interpolation without commutants","authors":"O. Szehr, R. Zarouf","doi":"10.7900/jot.2019may21.2264","DOIUrl":"https://doi.org/10.7900/jot.2019may21.2264","url":null,"abstract":"We introduce a ``dual-space approach'' to mixed Nevanlinna--Pick Carath'eodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting one of D. Sarason and B. Nagy-C. Foiac{s}. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This functional extension lemma plays a similar role as Sarason's commutant lifting theorem but it only involves the predual of X and no Hilbert space structure is needed. As an example, we present the respective Pick-type interpolation theorems for Beurling-Sobolev spaces.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49637329","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 : 2020-05-15DOI: 10.7900/jot.2018dec10.2246
Pan Ma, Fugang Yan, Dechao Zheng, Kehe Zhu
For entire functions f and g we determine exactly when the product H¯¯¯fT¯¯¯g of the Hankel operator H¯¯¯f and the Toeplitz operator T¯¯¯g is bounded on the Fock space F2α. This solves a natural companion to Sarason's Toeplitz product problem.
{"title":"Mixed products of Toeplitz and Hankel operators on the Fock space","authors":"Pan Ma, Fugang Yan, Dechao Zheng, Kehe Zhu","doi":"10.7900/jot.2018dec10.2246","DOIUrl":"https://doi.org/10.7900/jot.2018dec10.2246","url":null,"abstract":"For entire functions f and g we determine exactly when the product H¯¯¯fT¯¯¯g of the Hankel operator H¯¯¯f and the Toeplitz operator T¯¯¯g is bounded on the Fock space F2α. This solves a natural companion to Sarason's Toeplitz product problem.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43984700","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 : 2020-05-15DOI: 10.7900/JOT.2019JAN21.2230
Sara E. Arklint, James Gabe, Efren Ruiz
We show that a C∗-algebra A which is stably isomorphic to a unital graph C∗-algebra, is isomorphic to a graph C∗-algebra if and only if it admits an approximate unit of projections. As a consequence, a hereditary C∗-subalgebra of a unital real rank zero graph C∗-algebra is isomorphic to a graph C∗-algebra. Furthermore, if a C∗-algebra A admits an approximate unit of projections, then its minimal unitization is isomorphic to a graph C∗-algebra if and only if A is stably isomorphic to a unital graph C∗-algebra.
{"title":"Hereditary C∗-subalgebras of graph C∗-algebras","authors":"Sara E. Arklint, James Gabe, Efren Ruiz","doi":"10.7900/JOT.2019JAN21.2230","DOIUrl":"https://doi.org/10.7900/JOT.2019JAN21.2230","url":null,"abstract":"We show that a C∗-algebra A which is stably isomorphic to a unital graph C∗-algebra, is isomorphic to a graph C∗-algebra if and only if it admits an approximate unit of projections. As a consequence, a hereditary C∗-subalgebra of a unital real rank zero graph C∗-algebra is isomorphic to a graph C∗-algebra. Furthermore, if a C∗-algebra A admits an approximate unit of projections, then its minimal unitization is isomorphic to a graph C∗-algebra if and only if A is stably isomorphic to a unital graph C∗-algebra.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":"48 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71360544","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 : 2020-05-15DOI: 10.7900/jot.2019feb16.2258
Erik Christensen
Given two m×n matrices A=(aij) and B=(bij) with entries in B(H) for some Hilbert space H, the Schur block product is the m×n matrix A□B:=(aijbij). There exists an m×n matrix S=(sij) with entries from B(H) such that S is a contraction operator and A□B=(diag(AA∗))1/2S(diag(B∗B))1/2. The analogus result for the block Schur tensor product ⊠ defined by Horn and Mathias in cite{HM} holds too. This kind of decomposition of the Schur product seems to be unknown, even for scalar matrices. Based on the theory of random matrices we show that the set of contractions S, which may appear in such a decomposition, is a textit{thin} set in the ball of all contractions.
{"title":"Decompositions of block Schur product","authors":"Erik Christensen","doi":"10.7900/jot.2019feb16.2258","DOIUrl":"https://doi.org/10.7900/jot.2019feb16.2258","url":null,"abstract":"Given two m×n matrices A=(aij) and B=(bij) with entries in B(H) for some Hilbert space H, the Schur block product is the m×n matrix A□B:=(aijbij). There exists an m×n matrix S=(sij) with entries from B(H) such that S is a contraction operator and A□B=(diag(AA∗))1/2S(diag(B∗B))1/2. The analogus result for the block Schur tensor product ⊠ defined by Horn and Mathias in cite{HM} holds too. This kind of decomposition of the Schur product seems to be unknown, even for scalar matrices. Based on the theory of random matrices we show that the set of contractions S, which may appear in such a decomposition, is a textit{thin} set in the ball of all contractions.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46076883","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 : 2020-04-22DOI: 10.7900/jot.2020aug19.2305
M. Nagisa, Y. Watatani
e study several classes of general non-linear positive maps between C∗-algebras, which are not necessary completely positive maps. We characterize the class of the compositions of ∗-multiplicative maps and positive linear maps as the class of non-linear maps of boundedly positive type abstractly. We consider three classes of non-linear positive maps defined only on the positive cones, which are the classes of being monotone, supercongruent or concave. Any concave maps are monotone. The intersection of the monotone maps and the supercongruent maps characterizes the class of monotone Borel functional calculus. We give many examples of non-linear positive maps, which show that there exist no other relations among these three classes in general.
{"title":"Non-linear monotone positive maps","authors":"M. Nagisa, Y. Watatani","doi":"10.7900/jot.2020aug19.2305","DOIUrl":"https://doi.org/10.7900/jot.2020aug19.2305","url":null,"abstract":"e study several classes of general non-linear positive maps between C∗-algebras, which are not necessary completely positive maps. We characterize the class of the compositions of ∗-multiplicative maps and positive linear maps as the class of non-linear maps of boundedly positive type abstractly. We consider three classes of non-linear positive maps defined only on the positive cones, which are the classes of being monotone, supercongruent or concave. Any concave maps are monotone. The intersection of the monotone maps and the supercongruent maps characterizes the class of monotone Borel functional calculus. We give many examples of non-linear positive maps, which show that there exist no other relations among these three classes in general.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46277826","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 : 2020-04-16DOI: 10.7900/jot.2020jun25.2303
J. Galindo, E. Jord'a
Let G be a locally compact group and μ be a measure on G. In this paper we find conditions for the convolution operators λp(μ):Lp(G)→Lp(G) to be mean ergodic and uniformly mean ergodic. The ergodic properties of the operators λp(μ) are related to the ergodic properties of the measure μ as well.
{"title":"Ergodic properties of convolution operators","authors":"J. Galindo, E. Jord'a","doi":"10.7900/jot.2020jun25.2303","DOIUrl":"https://doi.org/10.7900/jot.2020jun25.2303","url":null,"abstract":"Let G be a locally compact group and μ be a measure on G. In this paper we find conditions for the convolution operators λp(μ):Lp(G)→Lp(G) to be mean ergodic and uniformly mean ergodic. The ergodic properties of the operators λp(μ) are related to the ergodic properties of the measure μ as well.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48809930","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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