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2012 | 60 | 2 | 133-154
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Infinite Asymptotic Games and (*)-Embeddings of Banach Spaces

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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 $G_{δ}$ 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.
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  • 14, Bizaniou Str., 164 51 Argiroupoli, Athens, Greece
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bwmeta1.element.bwnjournal-article-doi-10_4064-ba60-2-4
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