Let A be a unital C∗-algebra. We consider Jordan ∗-homomorphisms on C(X,A) and Jordan ∗-homomorphisms on Lip(X,A). More precisely, for any unital C∗-algebra A, we prove that every Jordan ∗-homomorphism on C(X,A) and every Jordan ∗-homomorphism on Lip(X,A) is represented as a weighted composition operator by using the irreducible representations of A. In addition, when A1 and A2 are primitive C∗-algebras, we characterize the Jordan ∗-isomorphisms. These results unify and enrich previous works on algebra ∗-homomorphisms on C(X,A) and Lip(X,A) for several concrete examples of A.
{"title":"Jordan $*$-homomorphisms on the spaces of continuous maps taking values in $C^*$-algebras","authors":"Shiho Oi","doi":"10.4064/sm220210-19-6","DOIUrl":"https://doi.org/10.4064/sm220210-19-6","url":null,"abstract":"Let A be a unital C∗-algebra. We consider Jordan ∗-homomorphisms on C(X,A) and Jordan ∗-homomorphisms on Lip(X,A). More precisely, for any unital C∗-algebra A, we prove that every Jordan ∗-homomorphism on C(X,A) and every Jordan ∗-homomorphism on Lip(X,A) is represented as a weighted composition operator by using the irreducible representations of A. In addition, when A1 and A2 are primitive C∗-algebras, we characterize the Jordan ∗-isomorphisms. These results unify and enrich previous works on algebra ∗-homomorphisms on C(X,A) and Lip(X,A) for several concrete examples of A.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47605999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study meromorphic functions on a domain $Omega subset mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a sequence of meromorphic functions is naturally defined on $Omega$ union a tree of spheres. In the second part, we show that a set $E subset Omega$ is removable if and only if it is negligible for extremal distance.
{"title":"On meromorphic functions whose image has finite spherical area","authors":"O. Ivrii","doi":"10.4064/sm220618-19-3","DOIUrl":"https://doi.org/10.4064/sm220618-19-3","url":null,"abstract":"In this paper, we study meromorphic functions on a domain $Omega subset mathbb{C}$ whose image has finite spherical area, counted with multiplicity. The paper is composed of two parts. In the first part, we show that the limit of a sequence of meromorphic functions is naturally defined on $Omega$ union a tree of spheres. In the second part, we show that a set $E subset Omega$ is removable if and only if it is negligible for extremal distance.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42514080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Hajl asz-Sobolev spaces, Hajl asz-Besov and Hajl asz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of the inhomogeneous version is new even when the underlying space is an RD-space.
{"title":"Pointwise characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type","authors":"Ryan Alvarado, Fan Wang, Dachun Yang, Wen Yuan","doi":"10.4064/sm210621-29-4","DOIUrl":"https://doi.org/10.4064/sm210621-29-4","url":null,"abstract":"In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Hajl asz-Sobolev spaces, Hajl asz-Besov and Hajl asz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of the inhomogeneous version is new even when the underlying space is an RD-space.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41726826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ ε )-injective for every ε > 0, yet is is not 2-injective and remarked in a footnote that Pe lczy´nski had proved for every λ > 1 the existence of a ( λ + ε )-injective space ( ε > 0) that is not λ injective. Unfortunately, no trace of the proof of Pe lczy´nski’s result has been preserved. In the present paper, we establish the said theorem for λ ∈ (1 , 2] by constructing an appropriate renorming of ℓ ∞ . This contrasts (at least for real scalars) with the case λ = 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.
{"title":"A forgotten theorem of Pełczyński: $(lambda +)$-injective spaces need not be $lambda $-injective—the case $lambda in (1,2]$","authors":"Tomasz Kania, G. Lewicki","doi":"10.4064/sm220119-25-6","DOIUrl":"https://doi.org/10.4064/sm220119-25-6","url":null,"abstract":". Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional 1-injective Banach space contains a hyperplane that is (2+ ε )-injective for every ε > 0, yet is is not 2-injective and remarked in a footnote that Pe lczy´nski had proved for every λ > 1 the existence of a ( λ + ε )-injective space ( ε > 0) that is not λ injective. Unfortunately, no trace of the proof of Pe lczy´nski’s result has been preserved. In the present paper, we establish the said theorem for λ ∈ (1 , 2] by constructing an appropriate renorming of ℓ ∞ . This contrasts (at least for real scalars) with the case λ = 1 for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45922559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. It is shown that for n ≤ 3 the joint numerical range of a family of commuting n × n complex matrices is always convex; for n ≥ 4 there are two commuting matrices whose joint numerical range is not convex.
． 那是展示用的n≤3《联合numerical太阳城of a family of通勤是n×n情结matrices总是凸;为n≥4有两条通勤matrices一个关节numerical太阳城是凸音符。
{"title":"The joint numerical range of commuting matrices","authors":"Pan-shun Lau, Chi-Kwong Li, Y. Poon","doi":"10.4064/sm200622-31-5","DOIUrl":"https://doi.org/10.4064/sm200622-31-5","url":null,"abstract":". It is shown that for n ≤ 3 the joint numerical range of a family of commuting n × n complex matrices is always convex; for n ≥ 4 there are two commuting matrices whose joint numerical range is not convex.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70508142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze the interpolation of the sum of a subspace, consisting of regular functions, with the span of a function with $r^alpha$-type singularity. In particular, we determine all interpolation parameters, for which the interpolation space of the subspace of regular functions is still a closed subspace. The main tool is here a result by Ivanov and Kalton on interpolation of subspaces. To apply it, we study the $K$-functional of the $r^alpha$-singular function. It turns out that the $K$-functional possesses upper and lower bounds that have a common decay rate at zero.
{"title":"Interpolation of a regular subspace complementing the span of a radially singular function","authors":"Konstantin Zerulla","doi":"10.5445/IR/1000134315","DOIUrl":"https://doi.org/10.5445/IR/1000134315","url":null,"abstract":"We analyze the interpolation of the sum of a subspace, consisting of regular functions, with the span of a function with $r^alpha$-type singularity. In particular, we determine all interpolation parameters, for which the interpolation space of the subspace of regular functions is still a closed subspace. The main tool is here a result by Ivanov and Kalton on interpolation of subspaces. To apply it, we study the $K$-functional of the $r^alpha$-singular function. It turns out that the $K$-functional possesses upper and lower bounds that have a common decay rate at zero.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70732696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weak limits of fractional Sobolev homeomorphisms\u0000are almost injective","authors":"A. Schikorra, J. Scott","doi":"10.4064/sm201218-20-9","DOIUrl":"https://doi.org/10.4064/sm201218-20-9","url":null,"abstract":"","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70508621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Integrability at infinity of negative powers of polynomials in the plane and its application to convergence of Dirichlet series","authors":"H. Hà, Thi-Thao Nguyen","doi":"10.4064/sm210211-19-11","DOIUrl":"https://doi.org/10.4064/sm210211-19-11","url":null,"abstract":"","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70515355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dual of the Choquet spaces with general Hausdorff content","authors":"Hiroki Saito, Hitoshi Tanaka","doi":"10.4064/sm210415-29-1","DOIUrl":"https://doi.org/10.4064/sm210415-29-1","url":null,"abstract":"","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70515853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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