Big slices of the unit ball in Banach spaces
Date
2020-07-10
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Abstract
Banachi ruumide geomeetrias on viimastel aastatel palju tähelepanu saanud diameeter-2 omadused, mis kirjeldavad selliseid Banachi ruume, kus ühikkera mistahes viilu diameeter on suurima võimaliku väärtusega ehk kaks. Selline omadus on näiteks klassikalistel Banachi ruumidel c0, ℓ∞, C[0,1], L1[0,1] ja L∞[0,1]. Refleksiivsed ruumid, erijuhul Hilberti ruumid, või separaablid kaasruumid, näiteks ℓ1, on Radon—Nikodými omadusega, mistõttu neis leidub kuitahes väikese läbimõõduga viile ja seega kõnealust omadust pole. Diameeter-2 omaduste suuna süstemaatilise uurimise käivitasid 2013. aastal T. A. Abrahamsen, V. Lima ja O. Nygaard. Käesoleva väitekirja põhieesmärk on süsteemselt uurida diameeter-2 omaduste tugevdusi ja nendega seotud mõisteid, nagu näiteks normi karedus ning Daugaveti tihkuse indeks. Töös kirjeldatakse täielikult ära erinevate tugevate diameeter-2 omaduste, karedate normide ja Daugaveti tihkuse indeksite stabiilsustulemused absoluutse normiga summaruumide ja komponentruumide vahel. Samuti tehakse kindlaks, millistel tingimustel vastavad omadused kanduvad ülemruumilt alamruumile ja vastupidi. Uuritakse tingimusi, mida meetriline ruum peab rahuldama, et vastaval Lipschitzi ruumil oleks *-nõrk sümmeetriline tugev diameeter-2 omadus. Töös üldistatakse kvantitatiivselt ka varasemalt teadaolevaid operaatorruumide kareduse tulemusi. Daugaveti tihkuse indeksi uurimise tulemusena vastatakse eitavalt Y. Ivakhno poolt 2006. aastal püstitatud küsimusele lokaalse diameeter-2 omaduse ja r-suurte viilude omaduse samaväärsuse kohta
Lately, in the field of the geometry of Banach spaces, a large amount of attention has been devoted to the study of diameter 2 properties which describe Banach spaces in which every slice of the unit ball has diameter 2. For example, the classical Banach spaces c0, ℓ∞, C[0,1], L1[0,1] and L∞[0,1] all have this property. On the other hand, in reflexive spaces, for example Hilbert spaces, or in separable dual spaces, for example ℓ1, there are slices with arbitrarily small diameter and thus they fail to have the diameter 2 properties. A systematic treatment of diameter 2 properties was started by T. A. Abrahamsen, V. Lima, and O. Nygaard in 2013. The aim of the thesis is to study strengthenings of the classical diameter-2 properties and related notions, such as roughness of the norm and the Daugavet indices of thickness. We give a complete description of how different versions of strong diameter 2 properties, rough norms and the Daugavet indices of thickness behave on absolute norms. The inheritance of these properties from a space to its subspaces, and vice versa, is also investigated. We study what criteria must a metric space meet in order for the space of Lipschitz functions on the metric space to have the w*-symmetric strong diameter 2 property. We quantitatively generalise known roughness results for spaces of bounded linear operators. As a result of our study of the Daugavet index of thickness, we give a negative answer to a question posed by Y. Ivakhno in 2006 about the relationship between the local diameter 2 property and the r-big slice property.
Lately, in the field of the geometry of Banach spaces, a large amount of attention has been devoted to the study of diameter 2 properties which describe Banach spaces in which every slice of the unit ball has diameter 2. For example, the classical Banach spaces c0, ℓ∞, C[0,1], L1[0,1] and L∞[0,1] all have this property. On the other hand, in reflexive spaces, for example Hilbert spaces, or in separable dual spaces, for example ℓ1, there are slices with arbitrarily small diameter and thus they fail to have the diameter 2 properties. A systematic treatment of diameter 2 properties was started by T. A. Abrahamsen, V. Lima, and O. Nygaard in 2013. The aim of the thesis is to study strengthenings of the classical diameter-2 properties and related notions, such as roughness of the norm and the Daugavet indices of thickness. We give a complete description of how different versions of strong diameter 2 properties, rough norms and the Daugavet indices of thickness behave on absolute norms. The inheritance of these properties from a space to its subspaces, and vice versa, is also investigated. We study what criteria must a metric space meet in order for the space of Lipschitz functions on the metric space to have the w*-symmetric strong diameter 2 property. We quantitatively generalise known roughness results for spaces of bounded linear operators. As a result of our study of the Daugavet index of thickness, we give a negative answer to a question posed by Y. Ivakhno in 2006 about the relationship between the local diameter 2 property and the r-big slice property.
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Banach spaces