Banach spaces without minimal subspaces – Examples
[Exemples d’espaces de Banach]
Ferenczi, Valentin ; Rosendal, Christian
Annales de l'Institut Fourier, Tome 62 (2012), p. 439-475 / Harvested from Numdam

Plusieurs exemples d’espaces de Banach séparables, dont certains sont nouveaux, sont analysés, et reliés à plusieurs dichotomies obtenues dans [11]. Ces exemples sont classifiés en fonction de quelle alternative de chaque dichotomie ils satisfont.

We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11],by classifying them according to which side of the dichotomies they fall.

Publié le : 2012-01-01
DOI : https://doi.org/10.5802/aif.2684
Classification:  46B03,  03E15
Mots clés: espaces de Banach étroits, dichotomies, classification des espaces de Banach
@article{AIF_2012__62_2_439_0,
     author = {Ferenczi, Valentin and Rosendal, Christian},
     title = {Banach spaces without minimal subspaces -- Examples},
     journal = {Annales de l'Institut Fourier},
     volume = {62},
     year = {2012},
     pages = {439-475},
     doi = {10.5802/aif.2684},
     zbl = {1254.46011},
     mrnumber = {2985506},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2012__62_2_439_0}
}
Ferenczi, Valentin; Rosendal, Christian. Banach spaces without minimal subspaces – Examples. Annales de l'Institut Fourier, Tome 62 (2012) pp. 439-475. doi : 10.5802/aif.2684. http://gdmltest.u-ga.fr/item/AIF_2012__62_2_439_0/

[1] Argyros, Spiros A.; Beanland, Kevin; Raikoftsalis, Theocharis An extremely non-homogeneous weak Hilbert space, Trans. Amer. Math. Soc. (to appear)

[2] Argyros, Spiros A.; Beanland, Kevin; Raikoftsalis, Theocharis A weak Hilbert space with few symmetries, C. R. Math. Acad. Sci. Paris, Tome 348 (2010) no. 23-24, pp. 1293-1296 | Article | MR 2745342

[3] Argyros, Spiros A.; Deliyanni, I. Examples of asymptotic l 1 Banach spaces, Trans. Amer. Math. Soc., Tome 349 (1997) no. 3, pp. 973-995 | Article | MR 1390965 | Zbl 0869.46002

[4] Argyros, Spiros A.; Deliyanni, I.; Kutzarova, D. N.; Manoussakis, A. Modified mixed Tsirelson spaces, J. Funct. Anal., Tome 159 (1998) no. 1, pp. 43-109 | Article | MR 1654174 | Zbl 0931.46017

[5] Argyros, Spiros A.; Haydon, R. A hereditarily indecomposable -space that solves the scalar-plus-compact problem, Acta Math., Tome 206 (2011) no. 1, pp. 1-54 | Article | MR 2784662

[6] Bossard, Benoît A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces, Fund. Math., Tome 172 (2002) no. 2, pp. 117-152 | Article | MR 1899225 | Zbl 0788.46007

[7] Casazza, Peter G. Some questions arising from the homogeneous Banach space problem, Banach spaces (Mérida, 1992), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 144 (1993), pp. 35-52 | MR 1209445 | Zbl 0805.46015

[8] Casazza, Peter G.; Shura, Thaddeus J. Tsirelson’s space, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1363 (1989) (With an appendix by J. Baker, O. Slotterbeck and R. Aron) | MR 981801 | Zbl 0709.46008

[9] Dilworth, S.; Ferenczi, V.; Kutzarova, D.; Odell, E. On strongly asymptotically p spaces and minimality, Journal of the London Math. Soc., Tome 75 (2007) no. 2, pp. 409-419 | Article | MR 2340235

[10] Ferenczi, Valentin; Rosendal, Christian Ergodic Banach spaces, Adv. Math., Tome 195 (2005) no. 1, pp. 259-282 | Article | MR 2145797

[11] Ferenczi, Valentin; Rosendal, Christian Banach spaces without minimal subspaces, J. Funct. Anal., Tome 257 (2009) no. 1, pp. 149-193 | Article | MR 2523338

[12] Gowers, W. T. A solution to Banach’s hyperplane problem, Bull. London Math. Soc., Tome 26 (1994) no. 6, pp. 523-530 | Article | MR 1315601 | Zbl 0838.46011

[13] Gowers, W. T. A hereditarily indecomposable space with an asymptotic unconditional basis, Geometric aspects of functional analysis (Israel, 1992–1994), Birkhäuser, Basel (Oper. Theory Adv. Appl.) Tome 77 (1995), pp. 112-120 | MR 1353454 | Zbl 0867.46007

[14] Gowers, W. T. A new dichotomy for Banach spaces, Geom. Funct. Anal., Tome 6 (1996) no. 6, pp. 1083-1093 | Article | MR 1421876 | Zbl 0868.46007

[15] Gowers, W. T. An infinite Ramsey theorem and some Banach-space dichotomies, Ann. of Math. (2), Tome 156 (2002) no. 3, pp. 797-833 | Article | MR 1954235

[16] Gowers, W. T.; Maurey, B. The unconditional basic sequence problem, J. Amer. Math. Soc., Tome 6 (1993) no. 4, pp. 851-874 | Article | MR 1201238 | Zbl 0827.46008

[17] Kutzarova, Denka; Leung, Denny H.; Manoussakis, Antonis; Tang, Wee-Kee Minimality properties of Tsirelson type spaces, Studia Math., Tome 187 (2008) no. 3, pp. 233-263 | Article | MR 2417456

[18] Manoussakis, A.; Pelczar, A. Quasi-minimality in mixed Tsirelson’s spaces, Math. Nachrichten (to appear)

[19] Schlumprecht, Thomas An arbitrarily distortable Banach space, Israel J. Math., Tome 76 (1991) no. 1-2, pp. 81-95 | Article | MR 1177333 | Zbl 0796.46007

[20] Tcaciuc, Adi On the existence of asymptotic-l p structures in Banach spaces, Canad. Math. Bull., Tome 50 (2007) no. 4, pp. 619-631 | Article | MR 2364212

[21] Tsirelson, B. S. Not every Banach space contains p or c 0 , Functional Anal. Appl., Tome 8 (1974), pp. 138-141 | Article | Zbl 0296.46018