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

Georgios-Nektarios I. Karadakis. "Infinite Asymptotic Games and (*)-Embeddings of Banach Spaces." Bulletin of the Polish Academy of Sciences. Mathematics 60.2 (2012): 133-154. <http://eudml.org/doc/281225>.

@article{Georgios2012,
abstract = {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.},
author = {Georgios-Nektarios I. Karadakis},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {functionally determined and strongly functionally determined sets; skipped-blocking decomposition; semi-embedding; -embedding},
language = {eng},
number = {2},
pages = {133-154},
title = {Infinite Asymptotic Games and (*)-Embeddings of Banach Spaces},
url = {http://eudml.org/doc/281225},
volume = {60},
year = {2012},
}

TY - JOUR
AU - Georgios-Nektarios I. Karadakis
TI - Infinite Asymptotic Games and (*)-Embeddings of Banach Spaces
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2012
VL - 60
IS - 2
SP - 133
EP - 154
AB - 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.
LA - eng
KW - functionally determined and strongly functionally determined sets; skipped-blocking decomposition; semi-embedding; -embedding
UR - http://eudml.org/doc/281225
ER -