We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a C*-algebra carrying a faithful tracial state is a Jordan homomorphism thus generalising Aupetit's 1998 result for finite von Neumann algebras.
{"title":"Invertibility preserving mappings onto finite $C^*$-algebras","authors":"Martin Mathieu, F. Schulz","doi":"10.4064/sm230101-27-3","DOIUrl":"https://doi.org/10.4064/sm230101-27-3","url":null,"abstract":"We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a C*-algebra carrying a faithful tracial state is a Jordan homomorphism thus generalising Aupetit's 1998 result for finite von Neumann algebras.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46695023","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}
Given a finite collection ${X_i}_{iin I}$ of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to $L^1$, we prove that their union has Lipschitz free space isomorphic to $L^1$. The short proof we provide is based on the Pelczy'nski decomposition method. A corollary is a solution to a question of Kaufmann about the union of two planar curves with tangential intersection. A second focus of the paper is on a special case of this result that can be studied using geometric methods. That is, we prove that the Lipschitz free space of a union of finitely many quasiconformal trees is isomorphic to $L^1$. These geometric methods also reveal that any metric quotient of a quasiconformal tree has Lipschitz free space isomorphic to $L^1$. Finally, we analyze Lipschitz light maps on unions and metric quotients of quasiconformal trees in order to prove that the Lipschitz dimension of any such union or quotient is equal to 1.
{"title":"Lipschitz functions on unions and quotients of metric spaces","authors":"D. Freeman, C. Gartland","doi":"10.4064/sm230117-19-4","DOIUrl":"https://doi.org/10.4064/sm230117-19-4","url":null,"abstract":"Given a finite collection ${X_i}_{iin I}$ of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to $L^1$, we prove that their union has Lipschitz free space isomorphic to $L^1$. The short proof we provide is based on the Pelczy'nski decomposition method. A corollary is a solution to a question of Kaufmann about the union of two planar curves with tangential intersection. A second focus of the paper is on a special case of this result that can be studied using geometric methods. That is, we prove that the Lipschitz free space of a union of finitely many quasiconformal trees is isomorphic to $L^1$. These geometric methods also reveal that any metric quotient of a quasiconformal tree has Lipschitz free space isomorphic to $L^1$. Finally, we analyze Lipschitz light maps on unions and metric quotients of quasiconformal trees in order to prove that the Lipschitz dimension of any such union or quotient is equal to 1.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44868299","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}
A nonnegative real function $f$ is said to be bell-shaped if it converges to zero at $pminfty$ and the $n$th derivative of $f$ changes sign $n$ times for every $n = 0, 1, 2, ldots$ In a similar way, we may say that a nonnegative sequence $a_k$ is bell-shaped if it converges to zero and the $n$th iterated difference of $a_k$ changes sign $n$ times for every $n = 0, 1, 2, ldots$ Bell-shaped functions were recently characterised by Thomas Simon and the first author. In the present paper we provide an analogous description of bell-shaped sequences. More precisely, we identify bell-shaped sequences with convolutions of P'olya frequency sequences and completely monotone sequences, and we characterise the corresponding generating functions as exponentials of appropriate Pick functions.
{"title":"Bell-shaped sequences","authors":"Mateusz Kwa'snicki, Jacek Wszola","doi":"10.4064/sm220923-2-2","DOIUrl":"https://doi.org/10.4064/sm220923-2-2","url":null,"abstract":"A nonnegative real function $f$ is said to be bell-shaped if it converges to zero at $pminfty$ and the $n$th derivative of $f$ changes sign $n$ times for every $n = 0, 1, 2, ldots$ In a similar way, we may say that a nonnegative sequence $a_k$ is bell-shaped if it converges to zero and the $n$th iterated difference of $a_k$ changes sign $n$ times for every $n = 0, 1, 2, ldots$ Bell-shaped functions were recently characterised by Thomas Simon and the first author. In the present paper we provide an analogous description of bell-shaped sequences. More precisely, we identify bell-shaped sequences with convolutions of P'olya frequency sequences and completely monotone sequences, and we characterise the corresponding generating functions as exponentials of appropriate Pick functions.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44144437","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 prove that convex functions on a $C_b(X)$ space satisfying a mild continuity condition can be represented using sigma additive measures. This generalises a result of Cheridito, Kupper and Tangpi,
{"title":"Convex increasing functionals on $C_b(X)$ spaces","authors":"F. Delbaen","doi":"10.4064/sm220927-26-1","DOIUrl":"https://doi.org/10.4064/sm220927-26-1","url":null,"abstract":"We prove that convex functions on a $C_b(X)$ space satisfying a mild continuity condition can be represented using sigma additive measures. This generalises a result of Cheridito, Kupper and Tangpi,","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49223888","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 work we study $S$-adic shifts generated by sequences of morphisms that are constant-length. We call a sequence of constant-length morphisms torsion-free if any prime divisor of one of the lengths is a divisor of infinitely many of the lengths. We show that torsion-free directive sequences generate shifts that enjoy the property of quasi-recognizability which can be used as a substitute for recognizability. Indeed quasi-recognizable directive sequences can be replaced by a recognizable directive sequence. With this, we give a finer description of the spectrum of shifts generated by torsion-free sequences defined on a sequence of alphabets of bounded size, in terms of extensions of the notions of height and column number. We illustrate our results throughout with examples that explain the subtleties that can arise.
{"title":"Torsion-free $S$-adic shifts and their spectrum","authors":"'Alvaro Bustos-Gajardo, Neil Mañibo, R. Yassawi","doi":"10.4064/sm221028-6-5","DOIUrl":"https://doi.org/10.4064/sm221028-6-5","url":null,"abstract":"In this work we study $S$-adic shifts generated by sequences of morphisms that are constant-length. We call a sequence of constant-length morphisms torsion-free if any prime divisor of one of the lengths is a divisor of infinitely many of the lengths. We show that torsion-free directive sequences generate shifts that enjoy the property of quasi-recognizability which can be used as a substitute for recognizability. Indeed quasi-recognizable directive sequences can be replaced by a recognizable directive sequence. With this, we give a finer description of the spectrum of shifts generated by torsion-free sequences defined on a sequence of alphabets of bounded size, in terms of extensions of the notions of height and column number. We illustrate our results throughout with examples that explain the subtleties that can arise.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45130206","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 a uniform algebra can have a nonzero bounded point derivation while having no nontrivial Gleason parts. Conversely, a uniform algebra can have a nontrivial Gleason part while having no nonzero, even possibly unbounded, point derivations.
{"title":"One-point Gleason parts and point derivations in uniform algebras","authors":"Swarup N. Ghosh, Alexander J. Izzo","doi":"10.4064/sm220729-12-12","DOIUrl":"https://doi.org/10.4064/sm220729-12-12","url":null,"abstract":"It is shown that a uniform algebra can have a nonzero bounded point derivation while having no nontrivial Gleason parts. Conversely, a uniform algebra can have a nontrivial Gleason part while having no nonzero, even possibly unbounded, point derivations.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43338276","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 define a monomial space to be a subspace of $ltwo$ that can be approximated by spaces that are spanned by monomial functions. We describe the structure of monomial spaces.
{"title":"Asymptotic Müntz–Szász theorems","authors":"J. Agler, John Mccarthy","doi":"10.4064/sm220713-19-2","DOIUrl":"https://doi.org/10.4064/sm220713-19-2","url":null,"abstract":"We define a monomial space to be a subspace of $ltwo$ that can be approximated by spaces that are spanned by monomial functions. We describe the structure of monomial spaces.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44010277","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}
. If X is an almost transitive Banach space with amenable isometry group (for example, if X = L p ([0 , 1]) with 1 < p < ∞ ) and X admits a uniformly continuous map X φ −→ E into a Banach space E satisfying inf k x − y k = r (cid:13)(cid:13) φ ( x ) − φ ( y ) (cid:13)(cid:13) > 0 for some r > 0 (that is, φ is almost uncollapsed ), then X admits a simultaneously uniform and coarse embedding into a Banach space V that is finitely representable in L 2 ( E ).
． 如果X是一个几乎传递巴拿赫空间与顺从的等距集团(例如,如果X = L p([0, 1])与1 < p <∞)和X承认均匀连续映射φ−→E为巴拿赫空间E满足正k X−y k = r (cid: 13) (cid: 13)φ(X)−φ(y) (cid: 13) (cid: 13) > 0对一些r > 0(φ几乎是气泡状),那么X承认同时制服和粗嵌入到巴拿赫空间V是在L有限表示的2 (E)。
{"title":"On uniform and coarse rigidity of $L^p([0,1])$","authors":"Christian Rosendal","doi":"10.4064/sm220603-6-8","DOIUrl":"https://doi.org/10.4064/sm220603-6-8","url":null,"abstract":". If X is an almost transitive Banach space with amenable isometry group (for example, if X = L p ([0 , 1]) with 1 < p < ∞ ) and X admits a uniformly continuous map X φ −→ E into a Banach space E satisfying inf k x − y k = r (cid:13)(cid:13) φ ( x ) − φ ( y ) (cid:13)(cid:13) > 0 for some r > 0 (that is, φ is almost uncollapsed ), then X admits a simultaneously uniform and coarse embedding into a Banach space V that is finitely representable in L 2 ( E ).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48659539","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}
This paper is devoted to theoretical aspects on optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of nonlinear $m$-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regards to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality parameters. Using an enhanced version of the original method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] for building almost greedy bases, we manage to construct bidemocratic bases whose unconditionality parameters satisfy significantly worse estimates than almost greedy bases even in Hilbert spaces.
{"title":"Sparse approximation using new greedy-like bases in superreflexive spaces","authors":"F. Albiac, J. L. Ansorena, M. Berasategui","doi":"10.4064/sm220506-3-2","DOIUrl":"https://doi.org/10.4064/sm220506-3-2","url":null,"abstract":"This paper is devoted to theoretical aspects on optimality of sparse approximation. We undertake a quantitative study of new types of greedy-like bases that have recently arisen in the context of nonlinear $m$-term approximation in Banach spaces as a generalization of the properties that characterize almost greedy bases, i.e., quasi-greediness and democracy. As a means to compare the efficiency of these new bases with already existing ones in regards to the implementation of the Thresholding Greedy Algorithm, we place emphasis on obtaining estimates for their sequence of unconditionality parameters. Using an enhanced version of the original method from [S. J. Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] for building almost greedy bases, we manage to construct bidemocratic bases whose unconditionality parameters satisfy significantly worse estimates than almost greedy bases even in Hilbert spaces.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44947346","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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