Genericity and amalgamation of classes of Banach spaces

By Spiros A. Argyros, Pandelis Dodos. 2007

48 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f163b527d547a750bcbabfa56130cff4fc23e4e

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In Applied Mathematics

  1. Christian Rosendal
  2. Alain Louveau
  3. V. Ferenczi

In Mathematical Physics

  1. V. Ferenczi
  2. Alain Louveau
  3. Christian Rosendal
  4. Lewis Bowen

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