Lifting bounded approximation properties from Banach spaces to their dual spaces
Kuupäev
2017-06-27
Autorid
Ajakirja pealkiri
Ajakirja ISSN
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Kirjastaja
Abstrakt
Aproksimatsiooniomadusi on uuritud alates 1930. aastatest. Süstemaatilised ja aktiivsed uuringud algasid 1955. aastal, mil Grothendieck oma kuulas memuaaris aproksimatsiooniomaduse mõiste kasutusele võttis. Aastal 2011 tõid Figiel, Johnson ja Pełczyński sisse uue tõkestatud aproksimatsiooniomaduse mõiste – Banachi ruumi ja tema kinnise alamruumi paari tõkestatud aproksimatsiooniomaduse. Hiljuti vaatlesid Figiel ja Johnson selle omaduse üldistust – Banachi ruumi kinniste alamruumide ahela tõkestatud aproksimatsiooniomadust. Käesoleva väitekirja põhieesmärk on süstemaatiliselt uurida paaride ja ahela tõkestatud aproksimatsiooniomadusi ning nende üldisemat versiooni – tõkestatud kumerat aproksimatsiooniomadust, mille tõid sisse Lissitsin ja Oja 2011 aastal. Viimane hõlmab erijuhul ka Banachi võrede positiivse aproksimatsiooniomaduse mõistet. Väitekirjas tõestatakse, et sellise paari tõkestatud aproksimatsiooniomadus, kus paar koosneb kaasruumist ja alamruumi annulaatorist, toob endaga kaasa vastava duaalse omaduse lähtepaari jaoks. Antud tulemust laiendatakse ka tõkestatud ahela aproksimatsiooniomaduste konteksti. Seejuures töötatakse välja uued lokaalse refleksiivsuse printsiibi versioonid, mis on kooskõlas Banachi ruumi kinniste alamruumide ahelatega. Töös leitakse tõkestatud kumera aproksimatsiooniomaduse efektiivne kriteerium, mille rakendusena tõestatakse üldistus Godefroy– Saphari teoreemile meetrilise aproksimatsiooniomaduse ülekandumiseks Banachi ruumi kaasruumile, mis rakendub ka Banachi võredes. Näidatakse, et tõkestatud kumerat aproksimatsiooniomadust saab lähteruumilt üle kanda kaasruumile kahel põhilisel juhul: 1) eeldusel, et lähteruum rahuldab laiendatava lokaalse refleksiivsuse ning lokaalse refleksiivsuse printsiibi teatavaid nõrgendatud versioone; 2) eeldusel, et kaasruumil on juba olemas tõkestatud kumera aproksimatsiooniomaduse nõrgem versioon. Need tulemused annavad üldise meetodi erisuguste tõkestatud aproksimatsiooniomaduste ülekandmiseks lähteruumilt kaasruumile ning üldistavad ja parendavad teadaolevaid tulemusi klassikalise tõkestatud aproksimatsiooniomaduse kohta.
The approximation properties have been studied since 1930s. Systematic studies started in 1955 when the term of the approximation property was introduced by Grothendieck in his famous memoir. In 2011, Figiel, Johnson, and Pełczyński introduced a new version of the bounded approximation property – the bounded approximation property for the pairs consisting of a Banach space and its closed subspace. Its extension – the bounded nest approximation property, involving nests of closed subspaces of a Banach space, was considered in 2016 by Figiel and Johnson. The main aim of the thesis is to systematically study the bounded approximation property of pairs, the bounded nest approximation property, and even more general version of the bounded approximation properties – the bounded convex approximation property launched by Lissitsin and Oja in 2011. The latter concept also includes the bounded positive approximation property of Banach lattices. In the thesis, it is proved that the bounded approximation property of a pair, consisting of a dual space and the annihilator of a closed subspace, implies the corresponding duality property for the pair of the Banach space and the closed subspace. The result is also extended to the context of the bounded nest approximation properties. For that, some new forms of the principle of local reflexivity are established. These forms respect given nests of subspaces. An effective characterization is developed for the bounded convex approximation property. As its application, an extension is obtained to the Godefroy–Saphar theorem on the lifting of the metric approximation property from a Banach space to its dual space so that it could be applied to the Banach lattices. It is shown that for the lifting of bounded convex approximation properties from a Banach space to its dual space rather weak forms of the principle of local reflexivity and the extendable of local reflexivity are sufficient. It is also shown that such a lifting is possible whenever the dual space already enjoys a weaker bounded convex approximation property. These results provide a unified approach to the lifting of various bounded approximation properties, extend and refine known theorems on classical bounded approximation properties.
The approximation properties have been studied since 1930s. Systematic studies started in 1955 when the term of the approximation property was introduced by Grothendieck in his famous memoir. In 2011, Figiel, Johnson, and Pełczyński introduced a new version of the bounded approximation property – the bounded approximation property for the pairs consisting of a Banach space and its closed subspace. Its extension – the bounded nest approximation property, involving nests of closed subspaces of a Banach space, was considered in 2016 by Figiel and Johnson. The main aim of the thesis is to systematically study the bounded approximation property of pairs, the bounded nest approximation property, and even more general version of the bounded approximation properties – the bounded convex approximation property launched by Lissitsin and Oja in 2011. The latter concept also includes the bounded positive approximation property of Banach lattices. In the thesis, it is proved that the bounded approximation property of a pair, consisting of a dual space and the annihilator of a closed subspace, implies the corresponding duality property for the pair of the Banach space and the closed subspace. The result is also extended to the context of the bounded nest approximation properties. For that, some new forms of the principle of local reflexivity are established. These forms respect given nests of subspaces. An effective characterization is developed for the bounded convex approximation property. As its application, an extension is obtained to the Godefroy–Saphar theorem on the lifting of the metric approximation property from a Banach space to its dual space so that it could be applied to the Banach lattices. It is shown that for the lifting of bounded convex approximation properties from a Banach space to its dual space rather weak forms of the principle of local reflexivity and the extendable of local reflexivity are sufficient. It is also shown that such a lifting is possible whenever the dual space already enjoys a weaker bounded convex approximation property. These results provide a unified approach to the lifting of various bounded approximation properties, extend and refine known theorems on classical bounded approximation properties.
Kirjeldus
Märksõnad
Banachi ruumid, lähendamine, Banach spaces, approximation