Genericity and amalgamation of classes of Banach spaces

Argyros, SA; Dodos, P

dc.contributor.author	Argyros, SA	en
dc.contributor.author	Dodos, P	en
dc.date.accessioned	2014-03-01T01:26:24Z
dc.date.available	2014-03-01T01:26:24Z
dc.date.issued	2007	en
dc.identifier.issn	0001-8708	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/18058
dc.subject	Banach spaces	en
dc.subject	Co-analytic ranks	en
dc.subject	Interpolation method	en
dc.subject	Schauder tree bases	en
dc.subject	Strong boundedness	en
dc.subject	Thin sets	en
dc.subject	Universal spaces	en
dc.subject.classification	Mathematics	en
dc.subject.other	COANALYTIC FAMILIES	en
dc.subject.other	BAIRE-1 FUNCTIONS	en
dc.subject.other	UNIVERSAL	en
dc.title	Genericity and amalgamation of classes of Banach spaces	en
heal.type	journalArticle	en
heal.identifier.primary	10.1016/j.aim.2006.05.013	en
heal.identifier.secondary	http://dx.doi.org/10.1016/j.aim.2006.05.013	en
heal.language	English	en
heal.publicationDate	2007	en
heal.abstract	We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C ([0, 1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L-1 (0, 1) such that the indices beta and r(ND) are unbounded on the set of Baire-1 elements of the ball of the double dual R** of R. This answers two questions of H.P. Rosenthal. We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y is an element of C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X. (c) 2006 Elsevier Inc. All rights reserved.	en
heal.publisher	ACADEMIC PRESS INC ELSEVIER SCIENCE	en
heal.journalName	Advances in Mathematics	en
dc.identifier.doi	10.1016/j.aim.2006.05.013	en
dc.identifier.isi	ISI:000244704600008	en
dc.identifier.volume	209	en
dc.identifier.issue	2	en
dc.identifier.spage	666	en
dc.identifier.epage	748	en