Singular actions on C*-algebras are automorphic group actions on C*-algebras, where the group need not be locally compact, or the action need not be strongly continuous. We study the covariant representation theory of such actions. In the usual case of strongly continuous actions of locally compact groups on C*-algebras, this is done via crossed products, but this approach is not available for singular C*-actions (this was our path in a previous paper). The literature regarding covariant representations for singular actions is already large and scattered, and in need of some consolidation. We collect in this survey a range of results in this field, mostly known. We improve some proofs and elucidate some interconnections. These include existence theorems by Borchers and Halpern, Arveson spectra, the Borchers-Arveson theorem, standard representations and Stinespring dilations as well as ground states, KMS states and ergodic states and the spatial structure of their GNS representations.
{"title":"Covariant representations for possibly singular actions on $C^*$-algebras","authors":"D. Beltiţă, H. Grundling, K. Neeb","doi":"10.4064/dm793-6-2019","DOIUrl":"https://doi.org/10.4064/dm793-6-2019","url":null,"abstract":"Singular actions on C*-algebras are automorphic group actions on C*-algebras, where the group need not be locally compact, or the action need not be strongly continuous. We study the covariant representation theory of such actions. In the usual case of strongly continuous actions of locally compact groups on C*-algebras, this is done via crossed products, but this approach is not available for singular C*-actions (this was our path in a previous paper). The literature regarding covariant representations for singular actions is already large and scattered, and in need of some consolidation. We collect in this survey a range of results in this field, mostly known. We improve some proofs and elucidate some interconnections. These include existence theorems by Borchers and Halpern, Arveson spectra, the Borchers-Arveson theorem, standard representations and Stinespring dilations as well as ground states, KMS states and ergodic states and the spatial structure of their GNS representations.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42694882","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}
H. Dales, N. Laustsen, T. Oikhberg, V. G. Troitsky
In 2012, Dales and Polyakov introduced the concepts of multi-norms and dual multi-norms based on a Banach space. Particular examples are the lattice multi-norm p} ̈ }Lnq and the dual lattice multi-norm p} ̈ } n q based on a Banach lattice. Here we extend these notions to cover ‘p–multi-norms’ for 1 ď p ď 8, where 8–multi-norms and 1–multi-norms correspond to multinorms and dual multi-norms, respectively. We shall prove two representation theorems. First we modify a theorem of Pisier to show that an arbitrary multi-normed space can be represented as ppY , } ̈ }Lnq : n P Nq, where Y is a closed subspace of a Banach lattice; we then give a version for certain p–multi-norms. Second, we obtain a dual version of this result, showing that an arbitrary dual multi-normed space can be represented as pppX{Y q, } ̈ } n q : n P Nq, where Y is a closed subspace of a Banach lattice X; again we give a version for certain p–multi-norms. We shall discuss several examples of p–multi-norms, including the weak p–summing norm and its dual and the canonical lattice p–multi-norm based on a Banach lattice. We shall determine the Banach spaces E such that the p–sum power-norm based on E is a p–multi-norm. This relies on a famous theorem of Kwapień; we shall present a simplified proof of this result. We shall relate p–multi-normed spaces to certain tensor products. Our representation theorems depend on the notion of ‘strong’ p–multi-norms, and we shall define these and discuss when p–multi-norms and strong p–multi-norms pass to subspaces, quotients, and duals; we shall also consider whether these multi-norms are preserved when we interpolate between couples of p–multi-normed spaces. We shall discuss multi-bounded operators between p–multi-normed spaces, and identify the classes of these spaces in some cases, in particular for spaces of operators between Banach lattices taken with their canonical lattice p–multi-norms. Acknowledgements. The authors are grateful to the London Mathematical Society for the award of Scheme 2 grant 21202 that allowed Troitsky to come to Lancaster in May 2013; to the Fields Institute in Toronto, for invitations to Dales, Laustsen, and Troitsky to participate in the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras in March and April, 2014; to the Lorentz Center in Leiden for invitations to Dales, Laustsen, and Troitsky to participate in a meeting on Ordered Banach Algebras in July, 2014. Oikhberg acknowledges with thanks the support of the Simons Foundation Travel Grant 210060, and Troitsky acknowledges with thanks the support of an NSERC grant. 2000 Mathematics Subject Classification: Primary 46B42, 46B20; secondary 46B28, 46B70.
{"title":"Multi-norms and Banach lattices","authors":"H. Dales, N. Laustsen, T. Oikhberg, V. G. Troitsky","doi":"10.4064/DM755-11-2016","DOIUrl":"https://doi.org/10.4064/DM755-11-2016","url":null,"abstract":"In 2012, Dales and Polyakov introduced the concepts of multi-norms and dual multi-norms based on a Banach space. Particular examples are the lattice multi-norm p} ̈ }Lnq and the dual lattice multi-norm p} ̈ } n q based on a Banach lattice. Here we extend these notions to cover ‘p–multi-norms’ for 1 ď p ď 8, where 8–multi-norms and 1–multi-norms correspond to multinorms and dual multi-norms, respectively. We shall prove two representation theorems. First we modify a theorem of Pisier to show that an arbitrary multi-normed space can be represented as ppY , } ̈ }Lnq : n P Nq, where Y is a closed subspace of a Banach lattice; we then give a version for certain p–multi-norms. Second, we obtain a dual version of this result, showing that an arbitrary dual multi-normed space can be represented as pppX{Y q, } ̈ } n q : n P Nq, where Y is a closed subspace of a Banach lattice X; again we give a version for certain p–multi-norms. We shall discuss several examples of p–multi-norms, including the weak p–summing norm and its dual and the canonical lattice p–multi-norm based on a Banach lattice. We shall determine the Banach spaces E such that the p–sum power-norm based on E is a p–multi-norm. This relies on a famous theorem of Kwapień; we shall present a simplified proof of this result. We shall relate p–multi-normed spaces to certain tensor products. Our representation theorems depend on the notion of ‘strong’ p–multi-norms, and we shall define these and discuss when p–multi-norms and strong p–multi-norms pass to subspaces, quotients, and duals; we shall also consider whether these multi-norms are preserved when we interpolate between couples of p–multi-normed spaces. We shall discuss multi-bounded operators between p–multi-normed spaces, and identify the classes of these spaces in some cases, in particular for spaces of operators between Banach lattices taken with their canonical lattice p–multi-norms. Acknowledgements. The authors are grateful to the London Mathematical Society for the award of Scheme 2 grant 21202 that allowed Troitsky to come to Lancaster in May 2013; to the Fields Institute in Toronto, for invitations to Dales, Laustsen, and Troitsky to participate in the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras in March and April, 2014; to the Lorentz Center in Leiden for invitations to Dales, Laustsen, and Troitsky to participate in a meeting on Ordered Banach Algebras in July, 2014. Oikhberg acknowledges with thanks the support of the Simons Foundation Travel Grant 210060, and Troitsky acknowledges with thanks the support of an NSERC grant. 2000 Mathematics Subject Classification: Primary 46B42, 46B20; secondary 46B28, 46B70.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45487147","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 first consider the additive Brownian motion process $(X(s_1,s_2), (s_1,s_2) in mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set ${(s_1,s_2)in mathbb{R}^2: X(s_1,s_2) >0}$ is equal to $$ frac{1}{4}left(1 + sqrt{13 + 4 sqrt{5}}right) simeq 1.421, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.
{"title":"Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet","authors":"R. Dalang, T. Mountford","doi":"10.4064/dm811-9-2021","DOIUrl":"https://doi.org/10.4064/dm811-9-2021","url":null,"abstract":"We first consider the additive Brownian motion process $(X(s_1,s_2), (s_1,s_2) in mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set ${(s_1,s_2)in mathbb{R}^2: X(s_1,s_2) >0}$ is equal to $$ \u0000frac{1}{4}left(1 + sqrt{13 + 4 sqrt{5}}right) simeq 1.421, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41845578","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 article unifies the theory for Hardy spaces built on Banach lattices on R-n satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy-Lorentz spaces, Hardy-Herz spaces, Hardy-Orlicz spaces, Hardy-Morrey spaces, Musielak-Orlicz-Hardy spaces, variable Hardy spaces and variable Hardy-Morrey spaces. Among them, Hardy-Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy-Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy-Littlewood maximal operator satisfies certain Fefferman-Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak-Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.
{"title":"Hardy spaces for ball quasi-Banach function spaces","authors":"Y. Sawano, K. Ho, Dachun Yang, Sibei Yang","doi":"10.4064/DM750-9-2016","DOIUrl":"https://doi.org/10.4064/DM750-9-2016","url":null,"abstract":"This article unifies the theory for Hardy spaces built on Banach lattices on R-n satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy-Lorentz spaces, Hardy-Herz spaces, Hardy-Orlicz spaces, Hardy-Morrey spaces, Musielak-Orlicz-Hardy spaces, variable Hardy spaces and variable Hardy-Morrey spaces. Among them, Hardy-Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy-Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy-Littlewood maximal operator satisfies certain Fefferman-Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak-Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157445","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":"Construction of some classes of nonlinear PDE’s admitting soliton solutions","authors":"A. Dellal, J. Henderson, A. Ouahab","doi":"10.4064/DM752-7-2016","DOIUrl":"https://doi.org/10.4064/DM752-7-2016","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157484","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}
$newcommand{mc}[1]{mathcal{#1}}$ $newcommand{D}{mc{D}(mc{Q},L^p,ell_w^q)}$ We present a framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces. Such a space $D$ is defined using a frequency covering $mc{Q}=(Q_i)_{iin I}$: If $(varphi_i)_{i}$ is a suitable partition of unity subordinate to $mc{Q}$, then $Vert gVert_{D}:=leftVertleft(Vertmc{F}^{-1}(varphi_ihat{g})Vert_{L^p}right)_{i}rightVert_{ell_w^q}$. We assume $mc{Q}=(T_iQ+b_i)_{i}$, with $T_iin{rm GL}(Bbb{R}^d),b_iinBbb{R}^d$. Given a prototype $gamma$, we consider the system [Psi_{c}=(L_{ccdot T_i^{-T}k}gamma^{[i]})_{iin I,kinBbb{Z}^d}text{ with }gamma^{[i]}=|det T_i|^{1/2}, M_{b_i}(gammacirc T_i^T),] with translation $L_x$ and modulation $M_{xi}$. We provide verifiable conditions on $gamma$ under which $Psi_c$ forms a Banach frame or an atomic decomposition for $D$, for small enough sampling density $c>0$. Our theory allows compactly supported prototypes and applies for arbitrary $p,qin(0,infty]$. Often, $Psi_c$ is both a Banach frame and an atomic decomposition, so that analysis sparsity is equivalent to synthesis sparsity, i.e. the analysis coefficients $(langle f,L_{ccdot T_i^{-T}k}gamma^{[i]}rangle)_{i,k}$ lie in $ell^p$ iff $f$ belongs to a certain decomposition space, iff $f=sum_{i,k}c_k^{(i)}cdot L_{ccdot T_i^{-T}k}gamma^{[i]}$ with $(c_k^{(i)})_{i,k}inell^p$. This is convenient if only analysis sparsity is known to hold: Generally, this only yields synthesis sparsity w.r.t. the dual frame, about which often only little is known. But our theory yields synthesis sparsity w.r.t. the well-understood primal frame. In particular, our theory applies to $alpha$-modulation spaces and inhom. Besov spaces. It also applies to shearlet frames, as we show in a companion paper.
{"title":"Structured, compactly supported Banach frame decompositions of decomposition spaces","authors":"F. Voigtlaender","doi":"10.4064/dm804-5-2021","DOIUrl":"https://doi.org/10.4064/dm804-5-2021","url":null,"abstract":"$newcommand{mc}[1]{mathcal{#1}}$ $newcommand{D}{mc{D}(mc{Q},L^p,ell_w^q)}$ We present a framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces. Such a space $D$ is defined using a frequency covering $mc{Q}=(Q_i)_{iin I}$: If $(varphi_i)_{i}$ is a suitable partition of unity subordinate to $mc{Q}$, then $Vert gVert_{D}:=leftVertleft(Vertmc{F}^{-1}(varphi_ihat{g})Vert_{L^p}right)_{i}rightVert_{ell_w^q}$. \u0000We assume $mc{Q}=(T_iQ+b_i)_{i}$, with $T_iin{rm GL}(Bbb{R}^d),b_iinBbb{R}^d$. Given a prototype $gamma$, we consider the system [Psi_{c}=(L_{ccdot T_i^{-T}k}gamma^{[i]})_{iin I,kinBbb{Z}^d}text{ with }gamma^{[i]}=|det T_i|^{1/2}, M_{b_i}(gammacirc T_i^T),] with translation $L_x$ and modulation $M_{xi}$. We provide verifiable conditions on $gamma$ under which $Psi_c$ forms a Banach frame or an atomic decomposition for $D$, for small enough sampling density $c>0$. Our theory allows compactly supported prototypes and applies for arbitrary $p,qin(0,infty]$. \u0000Often, $Psi_c$ is both a Banach frame and an atomic decomposition, so that analysis sparsity is equivalent to synthesis sparsity, i.e. the analysis coefficients $(langle f,L_{ccdot T_i^{-T}k}gamma^{[i]}rangle)_{i,k}$ lie in $ell^p$ iff $f$ belongs to a certain decomposition space, iff $f=sum_{i,k}c_k^{(i)}cdot L_{ccdot T_i^{-T}k}gamma^{[i]}$ with $(c_k^{(i)})_{i,k}inell^p$. This is convenient if only analysis sparsity is known to hold: Generally, this only yields synthesis sparsity w.r.t. the dual frame, about which often only little is known. But our theory yields synthesis sparsity w.r.t. the well-understood primal frame. \u0000In particular, our theory applies to $alpha$-modulation spaces and inhom. Besov spaces. It also applies to shearlet frames, as we show in a companion paper.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70158099","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}
Motivated by the quest for an analogue of the Gromov-Hausdorff distance in noncommutative geometry which is well-behaved with respect to C*-algebraic structures, we propose a complete metric on the class of Leibniz quantum compact metric spaces, named the dual Gromov-Hausdorff propinquity. This metric resolves several important issues raised by recent research in noncommutative metric geometry: it makes *-isomorphism a necessary condition for distance zero, it is well-adapted to Leibniz seminorms, and — very importantly — is complete, unlike the quantum propinquity which we introduced earlier. Thus our new metric provides a natural tool for noncommutative metric geometry, designed to allow for the generalizations of techniques from metric geometry to C*-algebra theory. Resume: Motives par la quete d’une metrique analogue a la distance de Gromov-Hausdorff pour la geometrie noncommutative et adaptee aux C*-algebres, nous proposons une distance complete sur la classe des espaces metriques compacts quantiques de Leibniz. Cette nouvelle distance, que nous appelons la proximite duale de Gromov-Hausdorff, resout plusieurs problemes importants que la recherche courante en geometrie metrique noncommutative a reveles. En particulier, il est necessaire pour les C*-algebres d’etre isomorphes pour avoir distance zero, et tous les espaces quantiques compacts impliques dans le calcul de la proximite duale sont de type Leibniz. En outre, notre distance est complete. Notre proximite duale de Gromov-Hausdorff est donc un nouvel outil naturel pour le developpement de la geometrie metrique noncommutative.
{"title":"The modular Gromov–Hausdorff propinquity","authors":"Frédéric Latrémolière","doi":"10.4064/dm778-5-2019","DOIUrl":"https://doi.org/10.4064/dm778-5-2019","url":null,"abstract":"Motivated by the quest for an analogue of the Gromov-Hausdorff distance in noncommutative geometry which is well-behaved with respect to C*-algebraic structures, we propose a complete metric on the class of Leibniz quantum compact metric spaces, named the dual Gromov-Hausdorff propinquity. This metric resolves several important issues raised by recent research in noncommutative metric geometry: it makes *-isomorphism a necessary condition for distance zero, it is well-adapted to Leibniz seminorms, and — very importantly — is complete, unlike the quantum propinquity which we introduced earlier. Thus our new metric provides a natural tool for noncommutative metric geometry, designed to allow for the generalizations of techniques from metric geometry to C*-algebra theory. Resume: Motives par la quete d’une metrique analogue a la distance de Gromov-Hausdorff pour la geometrie noncommutative et adaptee aux C*-algebres, nous proposons une distance complete sur la classe des espaces metriques compacts quantiques de Leibniz. Cette nouvelle distance, que nous appelons la proximite duale de Gromov-Hausdorff, resout plusieurs problemes importants que la recherche courante en geometrie metrique noncommutative a reveles. En particulier, il est necessaire pour les C*-algebres d’etre isomorphes pour avoir distance zero, et tous les espaces quantiques compacts impliques dans le calcul de la proximite duale sont de type Leibniz. En outre, notre distance est complete. Notre proximite duale de Gromov-Hausdorff est donc un nouvel outil naturel pour le developpement de la geometrie metrique noncommutative.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157914","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 partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_alpha)_{alphain P}$ of subsets of $Xtimes X$ such that $U_betasubset U_alpha$ for all $alphalebeta$ in $P$ and for every $xin X$ the family $(U_alpha[x])_{alphain P}$ of balls $U_alpha[x]={yin X:(x,y)in U_alpha}$ is a neighborhood base at $x$. A $P$-base $(U_alpha)_{alphain P}$ for $X$ is called locally uniform if the family of entourages $(U_alpha U_alpha^{-1}U_alpha)_{alphain P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $omega$-base. By Moore's Metrization Theorem, a topological space is metrizable if and only if it is a $T_0$-space with a locally uniform $omega$-base. In the paper we shall study topological spaces possessing a (locally uniform) $omega^omega$-base. Our results show that spaces with an $omega^omega$-base share some common properties with first countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight $omega^omega$-based topological spaces. On the other hand, topological spaces with a locally uniform $omega^omega$-base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $omega^omega$-base and show that such spaces are close to being $sigma$-compact.
{"title":"Topological spaces with an $omega^{omega}$-base","authors":"T. Banakh","doi":"10.4064/dm762-4-2018","DOIUrl":"https://doi.org/10.4064/dm762-4-2018","url":null,"abstract":"Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_alpha)_{alphain P}$ of subsets of $Xtimes X$ such that $U_betasubset U_alpha$ for all $alphalebeta$ in $P$ and for every $xin X$ the family $(U_alpha[x])_{alphain P}$ of balls $U_alpha[x]={yin X:(x,y)in U_alpha}$ is a neighborhood base at $x$. A $P$-base $(U_alpha)_{alphain P}$ for $X$ is called locally uniform if the family of entourages $(U_alpha U_alpha^{-1}U_alpha)_{alphain P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $omega$-base. By Moore's Metrization Theorem, a topological space is metrizable if and only if it is a $T_0$-space with a locally uniform $omega$-base. \u0000In the paper we shall study topological spaces possessing a (locally uniform) $omega^omega$-base. Our results show that spaces with an $omega^omega$-base share some common properties with first countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight $omega^omega$-based topological spaces. On the other hand, topological spaces with a locally uniform $omega^omega$-base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $omega^omega$-base and show that such spaces are close to being $sigma$-compact.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157661","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":"Hardy spaces with variable exponents on RD-spaces and applications","authors":"Ciqiang Zhuo, Y. Sawano, Dachun Yang","doi":"10.4064/DM744-9-2015","DOIUrl":"https://doi.org/10.4064/DM744-9-2015","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157592","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 completeness properties of the $L^1 $-space of a spectral measure","authors":"S. Okada, W. Ricker","doi":"10.4064/DM745-1-2016","DOIUrl":"https://doi.org/10.4064/DM745-1-2016","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157748","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：