Diameter two properties in spaces of Lipschitz functions
Date
2022-07-06
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Diameeter-2 omadused on Banachi ruumidel vaadeldavate teatud geomeetriliste omaduste koondnimetus. Sellise omadusega Banachi ruumi ühikkera kõik kindlat liiki osahulgad on diameetriga 2. Näiteks on Banachi ruumide l_∞, C[0, 1] ja L_1[0, 1] ühikkera iga viilu diameter 2. Teisalt aga leidub refleksiivsete ruumide või separaablite kaasruumide, nagu l_1, ühikkeras kuitahes väikese diameetriga viile.
Doktoriväitekirjas uuritakse diameeter-2 omadusi Lipschitzi funktsiooniruumides Lip_0(M), s.o. kõigi Lipschitzi funktsioonide f : M → R Banachi ruum, kus f norm on võrdne tema Lipschitzi konstandiga ja M on meetriline ruum. Ruum Lip_0(M) on kaasruum, tema teatud eelruum on Lipschitzi-vaba ruum F(M). Töö põhirõhk on diameeter-2 omadustel SSD2P ja w*-SSD2P. Väitekirjas antakse w*-SSD2P kirjeldus nii meetrilise ruumi M omadusena kui ka Lipschitzi-vaba ruumi F(M) omadusena. Viimast omadust uuritakse ka üldises Banachi ruumide kontekstis. Saadud kirjeldused sobituvad hästi teiste diameeter-2 omaduste kohta saadud varasemate kirjeldustega.
Töös saadakse de Leeuw’ teisenduse abil lihtsad piisavad tingimused ruumi Lip_0(M) diameeter-2 omaduste SSD2P, SD2P ja D2P jaoks, mis näitavad, et need omadused on Lipschitzi funktsiooniruumidel levinud, kuid siiski erinevad. Sellega parendatakse ja üldistatakse oluliselt mitmete matemaatikute tulemusi kitsaste erijuhtude jaoks. De Leeuw’ teisenduse võte on rakendatav laiemalt, näiteks lahendatakse selle abil hiljuti püstitatud küsimus ruumi Lip_0(M) Daugaveti punktide kohta.
Geometric properties of Banach spaces, which imply that all specific subsets of the unit ball have diameter 2, are usually referred to as diameter 2 properties. For example, every slice of the unit ball of Banach spaces l_∞, C[0, 1] and L_1[0, 1] has diameter equal to 2. On the other hand, the unit ball of a reflexive Banach space or a separable dual space like l_1 has slices of arbitrarily small diameter. In the thesis, we study diameter 2 properties for the Banach spaces Lip_0(M) of Lipschitz functions f : M → R, where the norm of f is its Lipschitz constant and M is a metric space. The space Lip_0(M) is a dual space with a specific predual, the Lipschitz-free space F(M). The emphasis of the work is on the diameter 2 properties SSD2P and w*-SSD2P. In the thesis, we characterise the w*-SSD2P for Lip_0(M) in terms of the metric space M and via a property of the Lipschitz-free space F(M). We also study the latter property in the general setting of Banach spaces. The given characterisations correspond well with previously obtained characterisations of other diameter 2 properties for the space Lip_0(M). By using the de Leeuw’s transform, we obtain simple sufficient conditions for the space Lip_0(M) to have the following diameter 2 properties: the SSD2P, the SD2P, and the D2P. Our results show that these diameter 2 properties are common but still different in the class of Lipschitz functions spaces. This significantly improves upon and generalises the previously obtained results for special cases by different mathematicians. The de Leeuw’s transform is applicable in a more general context. For instance, we use it to answer a recent question about Daugavet-points in the space Lip_0(M).
Geometric properties of Banach spaces, which imply that all specific subsets of the unit ball have diameter 2, are usually referred to as diameter 2 properties. For example, every slice of the unit ball of Banach spaces l_∞, C[0, 1] and L_1[0, 1] has diameter equal to 2. On the other hand, the unit ball of a reflexive Banach space or a separable dual space like l_1 has slices of arbitrarily small diameter. In the thesis, we study diameter 2 properties for the Banach spaces Lip_0(M) of Lipschitz functions f : M → R, where the norm of f is its Lipschitz constant and M is a metric space. The space Lip_0(M) is a dual space with a specific predual, the Lipschitz-free space F(M). The emphasis of the work is on the diameter 2 properties SSD2P and w*-SSD2P. In the thesis, we characterise the w*-SSD2P for Lip_0(M) in terms of the metric space M and via a property of the Lipschitz-free space F(M). We also study the latter property in the general setting of Banach spaces. The given characterisations correspond well with previously obtained characterisations of other diameter 2 properties for the space Lip_0(M). By using the de Leeuw’s transform, we obtain simple sufficient conditions for the space Lip_0(M) to have the following diameter 2 properties: the SSD2P, the SD2P, and the D2P. Our results show that these diameter 2 properties are common but still different in the class of Lipschitz functions spaces. This significantly improves upon and generalises the previously obtained results for special cases by different mathematicians. The de Leeuw’s transform is applicable in a more general context. For instance, we use it to answer a recent question about Daugavet-points in the space Lip_0(M).
Description
Keywords
Banach algebras, Banach spaces