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Genericity and amalgamation of classes of Banach spaces. (English) Zbl 1109.03047
Summary: 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 \({\mathcal C}\) 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 \({\mathcal C}\) of separable Banach spaces is strongly bounded if for every analytic subset \(A\) of \({\mathcal C}\) there exists \(Y\in{\mathcal C}\) 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 \(X\).

MSC:
03E15 Descriptive set theory
46B03 Isomorphic theory (including renorming) of Banach spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B70 Interpolation between normed linear spaces
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