The commuting bounded approximation property of Banach spaces
Date
2010-05-11T10:00:56Z
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Abstract
Väitekiri on funktsionaalanalüüsi alane uurimus. Funktsionaalanalüüs on matemaatikaharu, mille alguseks loetakse Banachi 1932. aastal ilmunud monograafiat lineaarsete operaatorite teooriast. Aproksimatsiooniomaduste (edaspidi a.o.) vallas on teinud 1950. aastatel põhjalikku tööd Grothendieck; ühele selle valdkonna põhiküsimusele (kas igal Banachi ruumil on a.o.?) andis (negatiivse) vastuse Enflo aastal 1972. Tänapäeval paeluvad a.o.-d paljusid Banachi ruumide uurijaid, kuna selles vallas on mitmeid pikka aega lahendamata jäänud probleeme.
Väitekirja põhieesmärk on uurida kommuteeruvat tõkestatud a.o.-t (ja selle kompaktset versiooni). Ühelt poolt on see omadus üldjuhul nõrgem kui kommuteeruv meetriline a.o. või lõplikumõõtmelise lahutuse omadus, teiselt poolt aga tugevam kui tõkestatud a.o.
Väitekirja teises peatükis tutvustatakse lugejale a.o.-te erinevaid versioone.
Kolmandas peatükis tõestatakse, et Willise poolt konstrueeritud meetriline kompaktne aproksimeeriv pere Willise ruumil XW on kommuteeruv, niisiis näidatakse, et kommuteeruv meetriline kompaktne a.o. ja a.o. on erinevad omadused.
Väitekirja neljandas peatükis parendatakse üht 1988. aastal Godefroy ja Saphari poolt saadud tulemust: näidatakse, kuidas Banachi ruumi X geomeetriline struktuur (M(a,B,c)-võrratus) võimaldab tõsta kommuteeruva tõkestatud kompaktse a.o. ruumilt X kaasruumi X*. Saadud tulemusele leitakse väitekirjas mitmeid rakendusi.
Viiendas peatükis keskendutakse ruumile XJS, mille konstrueerisid aastal 1996 Johnson ja Schechtman. Ruumil XJS ei ole meetrilist a.o.-t, kuid on, nagu tõestatakse väitekirjas, kommuteeruv 6-tõkestatud a.o.
Kuuendas peatükis võetakse kasutusele uus mõiste – asümptootiliselt kommuteeruv tõkestatud a.o. Näidatakse, et kui Banachi ruumil on asümptootiliselt kommuteeruv tõkestatud a.o., siis tal on separaabel lokaalse täiendatavuse omadus tugeval kujul.
This thesis is a study in functional analysis. Functional analysis is a branch of mathematics, the beginning of which is counted to be the monograph on the theory of linear operators due to Banach in 1932. Grothendieck has made thorough investigations on approximation properties (in the sequel: AP) in the 1950s; one of the core questions of the field (does every Banach space have the AP?) has been answered (in the negative) by Enflo in 1972. Nowadays the field of APs attracts many researchers of Banach spaces, since it contains a number of problems that have not been solved for a long time. The main aim of the thesis has been to investigate the commuting bounded AP (and also its compact version). On the one hand, the property is in general weaker than the commuting metric AP or the finite-dimensional decomposition property. On the other hand, it is stronger than the bounded AP. In Chapter 2 we make the reader familiar with several versions of APs. In Chapter 3 we prove that the metric compact approximation of the identity of the space XW due to Willis is commuting. This establishes that the commuting bounded compact AP and the AP are different properties. In Chapter 4 we extend a result by Godefroy and Saphar from 1988: we demonstrate how the geometric structure (satisfying the M(a,B,c)-inequality) of a Banach space permits to lift the commuting bounded compact AP to its dual space. We derive several applications from the result. Chapter 5 focuses on XJS, a space which was created in 1996 by Johnson and Schechtman. The space XJS fails the metric AP but has, as is proved in the thesis, the commuting 6-bounded AP. Chapter 6 coins a new term: the asymptotically commuting bounded AP. We prove that if a Banach space has the asymptotically commuting bounded AP, then it has a strong form of the separable local complementation property.
This thesis is a study in functional analysis. Functional analysis is a branch of mathematics, the beginning of which is counted to be the monograph on the theory of linear operators due to Banach in 1932. Grothendieck has made thorough investigations on approximation properties (in the sequel: AP) in the 1950s; one of the core questions of the field (does every Banach space have the AP?) has been answered (in the negative) by Enflo in 1972. Nowadays the field of APs attracts many researchers of Banach spaces, since it contains a number of problems that have not been solved for a long time. The main aim of the thesis has been to investigate the commuting bounded AP (and also its compact version). On the one hand, the property is in general weaker than the commuting metric AP or the finite-dimensional decomposition property. On the other hand, it is stronger than the bounded AP. In Chapter 2 we make the reader familiar with several versions of APs. In Chapter 3 we prove that the metric compact approximation of the identity of the space XW due to Willis is commuting. This establishes that the commuting bounded compact AP and the AP are different properties. In Chapter 4 we extend a result by Godefroy and Saphar from 1988: we demonstrate how the geometric structure (satisfying the M(a,B,c)-inequality) of a Banach space permits to lift the commuting bounded compact AP to its dual space. We derive several applications from the result. Chapter 5 focuses on XJS, a space which was created in 1996 by Johnson and Schechtman. The space XJS fails the metric AP but has, as is proved in the thesis, the commuting 6-bounded AP. Chapter 6 coins a new term: the asymptotically commuting bounded AP. We prove that if a Banach space has the asymptotically commuting bounded AP, then it has a strong form of the separable local complementation property.