no. 4
Infinite asymptotic games
[Jeux asymptotiques infinis]
Rosendal, Christian1
1 University of Illinois at Chicago Department of Mathematics, Statistics and Computer Science 322 Science and Engineering Offices (M/C 249) 851 S. Morgan Street Chicago, IL 60607-7045 (USA)
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 ℓ 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 ℓ 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.

DOI : 10.5802/aif.2467
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

Rosendal, Christian 1

1 University of Illinois at Chicago Department of Mathematics, Statistics and Computer Science 322 Science and Engineering Offices (M/C 249) 851 S. Morgan Street Chicago, IL 60607-7045 (USA) 
@article{AIF_2009__59_4_1359_0,
     author = {Rosendal, Christian},
     title = {Infinite asymptotic games},
     journal = {Annales de l'Institut Fourier},
     pages = {1359--1384},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {4},
     year = {2009},
     doi = {10.5802/aif.2467},
     mrnumber = {2566964},
     zbl = {1187.46006},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2467/}
}
TY  - JOUR
AU  - Rosendal, Christian
TI  - Infinite asymptotic games
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 1359
EP  - 1384
VL  - 59
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2467/
DO  - 10.5802/aif.2467
LA  - en
ID  - AIF_2009__59_4_1359_0
ER  - 
%0 Journal Article
%A Rosendal, Christian
%T Infinite asymptotic games
%J Annales de l'Institut Fourier
%D 2009
%P 1359-1384
%V 59
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2467/
%R 10.5802/aif.2467
%G en
%F AIF_2009__59_4_1359_0
Rosendal, Christian. Infinite asymptotic games. Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1359-1384. doi : 10.5802/aif.2467. https://aif.centre-mersenne.org/articles/10.5802/aif.2467/

[1] Albiac, F.; Kalton, N. J. Topics in Banach space theory, Graduate Texts in Mathematics, 233, Springer, New York, 2006 | MR | Zbl

[2] Bagaria, J.; López-Abad, J. Weakly Ramsey sets in Banach spaces, Adv. Math., Volume 160 (2001) no. 2, pp. 33-174 | DOI | MR | Zbl

[3] Dutta, S.; Fonf, V. On tree characterizations of G δ -embeddings and some Banach spaces, Israel J. Math., Volume 167 (2008), pp. 27-48 | DOI | MR

[4] Ferenczi, V.; Pelczar, A. M.; Rosendal, C. On a question of Haskell P. Rosenthal concerning a characterization of c 0 and l p , Bull. London Math. Soc., Volume 36 (2004) no. 3, pp. 96-406 | DOI | MR | Zbl

[5] Ferenczi, V.; Rosendal, C. Banach spaces without minimal subspaces (to apear in Journal of Funtional Analysis)

[6] Ferenczi, V.; Rosendal, C. Ergodic Banach spaces, Adv. Math., Volume 195 (2005) no. 1, pp. 59-282 | DOI | MR | Zbl

[7] Gowers, W. T. An infinite Ramsey theorem and some Banach-space dichotomies, Ann. of Math. (2), Volume 156 (2002) no. 3, pp. 97-833 | DOI | MR | Zbl

[8] Gowers, W. T.; Maurey, B. The unconditional basic sequence problem, J. Amer. Math. Soc., Volume 6 (1993) no. 4, pp. 51-874 | DOI | MR | Zbl

[9] Johnson, W.B.; Rosenthal, H.P.; Zippin, M. On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math., Volume 9 (1971), pp. 488-506 | DOI | MR | Zbl

[10] Kalton, N. J. On subspaces of c 0 and extensions of operators into C(K)- spaces, Q. J. Math., Volume 52 (2001), pp. 312-328 | DOI | MR | Zbl

[11] Kechris, A. S. Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995 | MR | Zbl

[12] Martin, D. A. A simple proof that determinacy implies Lebesgue measurability, Rend. Sem. Mat. Univ. Politec. Torino, Volume 61 (2003) no. 4, pp. 393-397 (2004) | MR | Zbl

[13] Maurey, B.; Milman, V. D.; Tomczak-Jaegermann, N. Asymptotic infinite-dimensional theory of Banach spaces, Geometric aspects of functional analysis (Israel, 1992–1994) (Oper. Theory Adv. Appl.), Volume 77, Birkhäuser, Basel, 1995, pp. 149-175 | MR | Zbl

[14] Odell, E.; Schlumprecht, Th. Trees and branches in Banach spaces, Trans. Amer. Math. Soc., Volume 354 (2002) no. 10, pp. 4085-4108 | DOI | MR | Zbl

[15] Odell, E.; Schlumprecht, Th. Embedding into Banach spaces with finite dimensional decompositions, Rev. R. Acad. Cien Serie A Mat., Volume 100 (2006) no. 2, pp. 1-28 | MR | Zbl

[16] Odell, E.; Sclumprecht, Th.; Zsák, A. On the structure of asymptotic ℓ p spaces, Q. J. Math., Volume 59 (2008) no. 1, pp. 85-122 | DOI | MR | Zbl

[17] Rosendal, C. An exact Ramsey principle for block sequences (to appear in Collectanea Mathematica)

Cité par Sources :