Infinite asymptotic games

Rosendal, Christian

Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1359-1384.

Nous étudions les jeux asymptotiques infinis dans les espaces de Banach admettant une décomposition en somme de sous-espaces de dimension finie (FDD). Nous montrons que les jeux analytiques sont déterminés en caractérisant précisément les conditions pour les deux joueurs d’avoir une stratégie gagnante.

Ces résultats servent à caractériser les espaces réflexifs qui se plongent dans une somme ${\ell }_p$ d’espaces de dimension finie, étendant ainsi des résultats d’Odell et Schlumprecht. Ils servent également à étudier les différentes notions d’homogénéité de bases et d’espaces de Banach. Nos résultats sont liés à des questions sur la vitesse d’extraction de sous-suites d’une suite normalisée faiblement nulle.

We study infinite asymptotic games in Banach spaces with a finite-dimensional decomposition (F.D.D.) and prove that analytic games are determined by characterising precisely the conditions for the players to have winning strategies. These results are applied to characterise spaces embeddable into ${\ell }_p$ sums of finite dimensional spaces, extending results of Odell and Schlumprecht, and to study various notions of homogeneity of bases and Banach spaces. The results are related to questions of rapidity of subsequence extraction from normalised weakly null sequences.

Classification : 46B03, 03E15
Keywords: Infinite asymptotic games, extraction of subsequences, weakly null trees
Mot clés : jeux asymptotiques infinis, extraction de sous-suites, arbres faiblement nuls

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