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

Karadakis G. N. I.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 2

133-154

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 S of sequences of finite subsets of X and the S-game. We prove that if S satisfies some specific stability conditions, then Player I has a winning strategy in the S-game if and only if X has a skipped-blocking decomposition each of whose skipped-blockings belongs to S. 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 Gδ 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.