Genericity and amalgamation of classes of Banach spaces |
| |
Authors: | Spiros A Argyros |
| |
Institution: | National Technical University of Athens, Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece |
| |
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 L1(0,1) such that the indices β and rND 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∈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. |
| |
Keywords: | 03E15 46B03 46B15 46B70 |
本文献已被 ScienceDirect 等数据库收录! |
|