Genericity and amalgamation of classes of Banach spaces

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 β and r N D 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.