Pub Date : 2021-12-17DOI: 10.7146/math.scand.a-135771
M. Amini, Sajad Zavar
We associate a boundary $mathcal B_{pi ,u}$ to each covariant representation $(pi ,u,H)$ of a $C^*$-dynamical system $(G,A,alpha )$ and study the action of $G$ on $mathcal B_{pi ,u}$ and its amenability properties. We relate rigidity properties of traces on the associated crossed product $C^*$-algebra to faithfulness of the action of the group on this boundary.
{"title":"Topological boundaries of covariant representations","authors":"M. Amini, Sajad Zavar","doi":"10.7146/math.scand.a-135771","DOIUrl":"https://doi.org/10.7146/math.scand.a-135771","url":null,"abstract":"We associate a boundary $mathcal B_{pi ,u}$ to each covariant representation $(pi ,u,H)$ of a $C^*$-dynamical system $(G,A,alpha )$ and study the action of $G$ on $mathcal B_{pi ,u}$ and its amenability properties. We relate rigidity properties of traces on the associated crossed product $C^*$-algebra to faithfulness of the action of the group on this boundary.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46206336","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 : 2021-12-08DOI: 10.7146/math.scand.a-134457
T. Brazelton, Stephen McKean
One can compute the local $mathbb{A}^1$-degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that for endomorphisms of the affine line, one can compute the local $mathbb{A}^1$-degree at points with inseparable residue field by taking a suitable lift of the polynomial and transferring its local degree. We also discuss the general set-up and strategy in terms of the six functor formalism. As an application, we show that trace forms of number fields are local $mathbb{A}^1$-degrees.
{"title":"Lifts, transfers, and degrees of univariate maps","authors":"T. Brazelton, Stephen McKean","doi":"10.7146/math.scand.a-134457","DOIUrl":"https://doi.org/10.7146/math.scand.a-134457","url":null,"abstract":"One can compute the local $mathbb{A}^1$-degree at points with separable residue field by base changing, working rationally, and post-composing with the field trace. We show that for endomorphisms of the affine line, one can compute the local $mathbb{A}^1$-degree at points with inseparable residue field by taking a suitable lift of the polynomial and transferring its local degree. We also discuss the general set-up and strategy in terms of the six functor formalism. As an application, we show that trace forms of number fields are local $mathbb{A}^1$-degrees.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47098113","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 : 2021-11-30DOI: 10.7146/math.scand.a-128963
Ibrahim Al-Ayyoub, M. Nasernejad, K. Khashyarmanesh, L. Roberts, V. C. Quiñonez
In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.
{"title":"Results on the normality of square-free monomial ideals and cover ideals under some graph operations","authors":"Ibrahim Al-Ayyoub, M. Nasernejad, K. Khashyarmanesh, L. Roberts, V. C. Quiñonez","doi":"10.7146/math.scand.a-128963","DOIUrl":"https://doi.org/10.7146/math.scand.a-128963","url":null,"abstract":"In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42951480","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 : 2021-11-30DOI: 10.7146/math.scand.a-128966
K. Ho
In this paper, we establish the extrapolation theory for the amalgam spaces and the Hardy-amalgam spaces. By using the extrapolation theory, we obtain the mapping properties for the Calderón-Zygmund operators and its commutator, the Carleson operators and establish the Rubio de Francia inequalities for Littlewood-Paley functions of arbitrary intervals to the amalgam spaces. We also obtain the boundedness of the Calder{ó}n-Zygmund operators and the intrinsic square function on the Hardy-amalgam spaces.
本文建立了汞齐空间和hardy汞齐空间的外推理论。利用外推理论，得到了Calderón-Zygmund算子及其交换子Carleson算子的映射性质，建立了任意区间的Littlewood-Paley函数到混合空间的Rubio de Francia不等式。得到了Hardy-amalgam空间上Calder{ó}n-Zygmund算子的有界性和内禀平方函数。
{"title":"Singular integrals and sublinear operators on amalgam spaces and Hardy-amalgam spaces","authors":"K. Ho","doi":"10.7146/math.scand.a-128966","DOIUrl":"https://doi.org/10.7146/math.scand.a-128966","url":null,"abstract":"In this paper, we establish the extrapolation theory for the amalgam spaces and the Hardy-amalgam spaces. By using the extrapolation theory, we obtain the mapping properties for the Calderón-Zygmund operators and its commutator, the Carleson operators and establish the Rubio de Francia inequalities for Littlewood-Paley functions of arbitrary intervals to the amalgam spaces. We also obtain the boundedness of the Calder{ó}n-Zygmund operators and the intrinsic square function on the Hardy-amalgam spaces.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44453123","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 : 2021-11-30DOI: 10.7146/math.scand.a-128968
Qingshan Zhou, Liulan Li, A. Rasila
Let $Omega subset mathbb{R}^n$ be a Gromov hyperbolic, $varphi$-length John domain. We show that there is a uniformly continuous identification between the inner boundary of $Omega$ and the Gromov boundary endowed with a visual metric, By using this result, we prove the boundary continuity not only for quasiconformal homeomorphisms, but also for more generally rough quasi-isometries between the domains equipped with the quasihyperbolic metrics.
{"title":"Generalized John Gromov hyperbolic domains and extensions of maps","authors":"Qingshan Zhou, Liulan Li, A. Rasila","doi":"10.7146/math.scand.a-128968","DOIUrl":"https://doi.org/10.7146/math.scand.a-128968","url":null,"abstract":"Let $Omega subset mathbb{R}^n$ be a Gromov hyperbolic, $varphi$-length John domain. We show that there is a uniformly continuous identification between the inner boundary of $Omega$ and the Gromov boundary endowed with a visual metric, By using this result, we prove the boundary continuity not only for quasiconformal homeomorphisms, but also for more generally rough quasi-isometries between the domains equipped with the quasihyperbolic metrics.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47722906","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 : 2021-11-30DOI: 10.7146/math.scand.a-129007
N. Q. Dieu, T. V. Long
In this note, we establish a product property for $P$-extremal functions in the same spirit as the original product formula due to J. Siciak in Ann. Polon. Math., 39 (1981), 175–211. As a consequence, we obtain convexity for the sublevel sets of such extremal functions. Moreover, we also generalize the product property of $P$-extremal functions established by L. Bos and N. Levenberg in Comput. Methods Funct. Theory 18 (2018), 361–388, and later by N. Levenberg and M. Perera, in Contemporary Mathematics 743 (2020), 11–19, in which no restriction on $P$ is needed.
{"title":"Product property of global $P$-extremal functions","authors":"N. Q. Dieu, T. V. Long","doi":"10.7146/math.scand.a-129007","DOIUrl":"https://doi.org/10.7146/math.scand.a-129007","url":null,"abstract":"In this note, we establish a product property for $P$-extremal functions in the same spirit as the original product formula due to J. Siciak in Ann. Polon. Math., 39 (1981), 175–211. As a consequence, we obtain convexity for the sublevel sets of such extremal functions. Moreover, we also generalize the product property of $P$-extremal functions established by L. Bos and N. Levenberg in Comput. Methods Funct. Theory 18 (2018), 361–388, and later by N. Levenberg and M. Perera, in Contemporary Mathematics 743 (2020), 11–19, in which no restriction on $P$ is needed.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45902559","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 : 2021-11-30DOI: 10.7146/math.scand.a-128541
Marek Golasi´nski
Let $mathbb{O}P^2_{(p)}$ be the $p$-localization of the Cayley projective plane $mathbb{O}P^2$ for a prime $p$ or $p=0$. We show that the homotopy nilpotency class $textrm{nil} Omega(mathbb{O}P^2_{(p)})2$ and $textrm{nil} Omega (mathbb{O}P^2_{(p)})=1$ for $p>5$ or $p=0$. The homotopy nilpotency of remaining Rosenfeld projective planes are discussed as well.
{"title":"On homotopy nilpotency of the octonian plane $mathbb{O}P^2$","authors":"Marek Golasi´nski","doi":"10.7146/math.scand.a-128541","DOIUrl":"https://doi.org/10.7146/math.scand.a-128541","url":null,"abstract":"Let $mathbb{O}P^2_{(p)}$ be the $p$-localization of the Cayley projective plane $mathbb{O}P^2$ for a prime $p$ or $p=0$. We show that the homotopy nilpotency class $textrm{nil} Omega(mathbb{O}P^2_{(p)})<infty $ for $p>2$ and $textrm{nil} Omega (mathbb{O}P^2_{(p)})=1$ for $p>5$ or $p=0$. The homotopy nilpotency of remaining Rosenfeld projective planes are discussed as well.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46531093","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 : 2021-11-30DOI: 10.7146/math.scand.a-128969
Venuste Nyagahakwa, Gratien Haguma
In this paper, we prove that each topological group isomorphism of the additive topological group $(mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $mathbb{R}$ onto itself.
{"title":"Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups","authors":"Venuste Nyagahakwa, Gratien Haguma","doi":"10.7146/math.scand.a-128969","DOIUrl":"https://doi.org/10.7146/math.scand.a-128969","url":null,"abstract":"In this paper, we prove that each topological group isomorphism of the additive topological group $(mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $mathbb{R}$ onto itself.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41960400","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 : 2021-10-14DOI: 10.7146/math.scand.a-134348
Y. Choi, Mingu Jung
In this paper, we study boundaries for the Gelfand transform image of infinite dimensional analogues of the classical disk algebras. More precisely, given a certain Banach algebra $mathcal{A}$ of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$, we show that the Shilov boundary for the Gelfand transform image of $mathcal{A}$ can be explicitly described for a large class of Banach spaces. Some possible application of our result to the famous Corona theorem is also briefly discussed.
{"title":"Boundaries for Gelfand transform images of Banach algebras of holomorphic functions","authors":"Y. Choi, Mingu Jung","doi":"10.7146/math.scand.a-134348","DOIUrl":"https://doi.org/10.7146/math.scand.a-134348","url":null,"abstract":"In this paper, we study boundaries for the Gelfand transform image of infinite dimensional analogues of the classical disk algebras. More precisely, given a certain Banach algebra $mathcal{A}$ of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$, we show that the Shilov boundary for the Gelfand transform image of $mathcal{A}$ can be explicitly described for a large class of Banach spaces. Some possible application of our result to the famous Corona theorem is also briefly discussed.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43029750","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 : 2021-09-03DOI: 10.7146/math.scand.a-133419
M. Tsakiris, Sihang Xu
We pose and study the Fermat-Torricelli problem for a triangle in the projective plane under the sine distance. Our main finding is that if every side of the triangle has length greater than $sin 60^circ $, then the Fermat-Torricelli point is the vertex opposite the longest side. Our proof relies on a complete characterization of the equilateral case together with a deformation argument.
{"title":"The Fermat-Torricelli problem in the projective plane","authors":"M. Tsakiris, Sihang Xu","doi":"10.7146/math.scand.a-133419","DOIUrl":"https://doi.org/10.7146/math.scand.a-133419","url":null,"abstract":"We pose and study the Fermat-Torricelli problem for a triangle in the projective plane under the sine distance. Our main finding is that if every side of the triangle has length greater than $sin 60^circ $, then the Fermat-Torricelli point is the vertex opposite the longest side. Our proof relies on a complete characterization of the equilateral case together with a deformation argument.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48032848","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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