Infinite Asymptotic Games and (*)-Embeddings of Banach Spaces

Georgios-Nektarios I. Karadakis

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 60 (2012), p. 133-154 / Harvested from The Polish Digital Mathematics Library

Résumé

We use methods of infinite asymptotic games to characterize subspaces of Banach spaces with a finite-dimensional decomposition (FDD) and prove new theorems on operators. We consider a separable Banach space X, a set of sequences of finite subsets of X and the -game. We prove that if satisfies some specific stability conditions, then Player I has a winning strategy in the -game if and only if X has a skipped-blocking decomposition each of whose skipped-blockings belongs to . This result implies that if T is a (*)-embedding of X (a 1-1 operator which maps the balls of subspaces with an FDD to weakly $G_{δ}$ sets), then, for every n ≥ 4, there exist n subspaces of X with an FDD that generate X and the restriction of T to each of them is a semi-embedding under an equivalent norm. We also prove that X does not contain isomorphic copies of dual spaces if and only if every (*)-embedding defined on X is an isomorphic embedding. We also deal with the case where X is non-separable, reaching similar results.

Détail

Publié le : 2012-01-01

Zbl 1250.46006

EUDML-ID : urn:eudml:doc:281225