Advertisement

Mathematische Annalen

, Volume 333, Issue 4, pp 831–858 | Cite as

Operators on C[0,1] preserving copies of asymptotic ℓ1 spaces

  • I. GasparisEmail author
Article

Abstract

Given separable Banach spaces X, Y, Z and a bounded linear operator T:XY, then T is said to preserve a copy of Z provided that there exists a closed linear subspace E of X isomorphic to Z and such that the restriction of T to E is an into isomorphism. It is proved that every operator on C([0,1]) which preserves a copy of an asymptotic ℓ1 space also preserves a copy of C([0,1]).

Keywords

Operators on C(K) space asymptotic ℓ1 space weakly null sequence Schreier sets 

Mathematics Subject Classification (2000)

Primary: 46B03 Secondary: 06A07 03E02 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alspach, D.: Quotients of C[0,1] with separable dual. Israel J. Math. 29, 360–384 (1978)Google Scholar
  2. 2.
    Alspach, D.: C(K) norming subsets of C[0,1]. Studia Math. 70, 27–61 (1981)Google Scholar
  3. 3.
    Alspach, D.: Operators on Cα) which do not preserve Cα). Fund. Math. 153, 81–98 (1997)Google Scholar
  4. 4.
    Alspach, D., Argyros, S.A.: Complexity of weakly null sequences. Dissertationes Mathematicae, 321, 1–44 (1992)Google Scholar
  5. 5.
    Alspach, D., Benyamini, Y.: C(K) quotients of separable Open image in new window spaces. Israel J. Math. 32 145–160 (1979),Google Scholar
  6. 6.
    Argyros, S.A., Deliyanni, I.: Examples of asymptotic ℓ1 Banach spaces. Trans. Amer. Math. Soc. 349 973–995 (1997),Google Scholar
  7. 7.
    Argyros, S.A., Deliyanni, I., Kutzarova, D.N., Manoussakis, A.: Modified mixed Tsirelson spaces. J. Funct. Anal. 159(1), 43–109 (1998)CrossRefGoogle Scholar
  8. 8.
    Argyros, S.A., Gasparis, I.: Unconditional structures of weakly null sequences. Trans. Amer. Math. Soc. 353, 2019–2058 (2001)CrossRefGoogle Scholar
  9. 9.
    Argyros, S.A., Godefroy, G., Rosenthal, H.P.: Descriptive set theory and Banach spaces. Handbook of the geometry of Banach spaces, Vol.2, (W.B. Johnson and J. Lindenstrauss eds.), North Holland, 1007–1069 (2003)Google Scholar
  10. 10.
    Argyros, S.A., Mercourakis, S. Tsarpalias, A.: Convex unconditionality and summability of weakly null sequences. Israel J. Math. 107, 157–193 (1998)Google Scholar
  11. 11.
    Benyamini, Y.: An extension theorem for separable Banach spaces. Israel J. Math. 29, 24–30 (1978)Google Scholar
  12. 12.
    Bessaga, C., Pelczynski, A.: Spaces of continuous functions IV. Studia Math. 19, 53–62 (1960)Google Scholar
  13. 13.
    Bourgain, J.: A result on operators on C[0,1]. J. Operator Theory 3(2), 275–289 (1980)Google Scholar
  14. 14.
    Bourgain, J.: The Szlenk index and operators on C(K) spaces. Bull. Soc. Math. Belg. Ser. B 31, 87–117 (1979)Google Scholar
  15. 15.
    Ellentuck, E.: A new proof that analytic sets are Ramsey. J. Symbolic Logic 39, 163–165 (1974)Google Scholar
  16. 16.
    Elton, J.: Thesis. Yale University (1978)Google Scholar
  17. 17.
    Figiel, T., Ghoussoub, N., Johnson, W.B.: On the structure of non-weakly compact operators on Banach lattices. Math. Ann. 257, 317–334 (1981)CrossRefGoogle Scholar
  18. 18.
    Figiel, T., Johnson, W.B.: A uniformly convex Banach space which contains no ℓp. Compositio Math. 29, 179–190 (1974)Google Scholar
  19. 19.
    Gasparis, I. A dichotomy theorem for subsets of the power set of the natural numbers. Proc. Amer. Math. Soc. 129, 759–764 (2001)Google Scholar
  20. 20.
    Gasparis, I., Odell, E., Wahl, B.: Weakly null sequences in the Banach space C(K). preprintGoogle Scholar
  21. 21.
    Gowers, W., Maurey, B.: The unconditional basic sequence problem. J. Amer. Math. Soc. 6, 851–874 (1993)Google Scholar
  22. 22.
    Haydon, R.: An extreme point criterion for separability of a dual Banach space and a new proof of a theorem of Corson. Quart. J. Math. Oxford 27, 379–385 (1976)Google Scholar
  23. 23.
    Judd, R.: A dichotomy on Schreier sets. Studia Math. 132, 245–256 (1999)Google Scholar
  24. 24.
    Kutzarova, D., Lin, P.K.: Remarks about Schlumprecht space. Proc. Amer. Math. Soc. 128, 2059–2068 (2000)CrossRefGoogle Scholar
  25. 25.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces I. Springer-Verlag, New York (1977)Google Scholar
  26. 26.
    Lindenstrauss, J., Wulbert, D.E.: On the classification of the Banach spaces whose duals are L1 spaces. J. Funct. Anal. 4, 332–349 (1969)CrossRefGoogle Scholar
  27. 27.
    Maurey, B., Milman, V.D., Tomczak-Jaegermann, N.: Asymptotic infinite-dimensional theory of Banach spaces. Operator Theory: Adv. Appl. 77, 149–175 (1995)Google Scholar
  28. 28.
    Maurey, B., Pisier, G.: Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach. (French), Studia Math. 58, 45–90 (1976)Google Scholar
  29. 29.
    Mazurkiewicz, S., Sierpinski, W.: Contributions a la topologie des ensembles denomrables. Fund. Math. 1, 17–27 (1920)Google Scholar
  30. 30.
    Miljutin, A.: Isomorphism of the spaces of continuuous functions over compact sets of the cardinality of the continuum. (Russian), Teor. FunkciiFunkcional. Anal. i Priložen. Vyp 2, 150–156 (1966)Google Scholar
  31. 31.
    Milman, V.D., Tomczak-Jaegermann, N.: Asymptotic ℓp spaces and bounded distortion. Contemp. Math. 144, 173–195 (1993)Google Scholar
  32. 32.
    Odell, E.: Applications of Ramsey theorems to Banach space theory. Notes in Banach spaces, (H.E. Lacey, ed.), Univ. Texas Press, 379–404 (1980)Google Scholar
  33. 33.
    Odell, E.: On subspaces, asymptotic structure, and distortion of Banach spaces, connections with logic. Analysis and Logic (Mons, 1997), 189–267, London Math. Soc. Lecture Note Ser., 262 Cambridge University Press, Cambridge, 2002Google Scholar
  34. 34.
    Odell, E.: Ordinal indices in Banach spaces. Extracta Math., 19, 93–125 (2004)Google Scholar
  35. 35.
    Odell, E., Schlumprecht, T.: On the richness of the set of p's in Krivine's theorem. Oper. Theory Adv. Appl. 77, 177–198 (1995)Google Scholar
  36. 36.
    Odell, E., Schlumprecht, T.: A Banach space block finitely universal for monotone bases. Trans. Amer. Math. Soc. 352, 1859–1888 (2000)CrossRefGoogle Scholar
  37. 37.
    Odell, E., Tomczak-Jaegermann, N., Wagner, R.: Proximity to ℓ1 and distortion in asymptotic ℓ1 spaces. J. Funct. Anal. 150, 101–145 (1997)CrossRefGoogle Scholar
  38. 38.
    Pelczynski, A.: Projections in certain Banach spaces. Studia Math. 19, 209–228 (1960)Google Scholar
  39. 39.
    Pelczynski, A.: Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuuous functions. Dissertationes Math. Rozpraway Mat. 58 (1968)Google Scholar
  40. 40.
    Pelszynski, A.: On C(S)-subspaces of separable Banach spaces. Studia Math. 31, 513–522 (1968)Google Scholar
  41. 41.
    Pelczynski, A., Semadeni, Z.: Spaces of continuous functions III. Spaces C(Ω) for Ω without perfect subsets. Studia Math. 18, 211–222 (1959)Google Scholar
  42. 42.
    Rosenthal, H.P.: On factors of C[0,1] with non-separable dual. Israel J. Math. 13, 361–378 (1972)Google Scholar
  43. 43.
    Rosenthal, H.P.: A characterization of Banach spaces containing ℓ1. Proc. Nat. Acad. Sci. (USA) 71, 2411–2413 (1974)Google Scholar
  44. 44.
    Rosenthal, H.P.: The Banach spaces C(K). Handbook of the geometry of Banach spaces, Vol.2, (W.B. Johnson and J. Lindenstrauss eds.), North Holland, 1547–1602 (2003)Google Scholar
  45. 45.
    Schlumprecht, T.: An arbitrarily distortable Banach space. Israel J. Math. 76, 81–95 (1991)Google Scholar
  46. 46.
    Schreier, J.: Ein Gegenbeispiel zur theorie der schwachen konvergenz. Studia Math. 2, 58–62 (1930)Google Scholar
  47. 47.
    Szlenk, W.: The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces. Studia Math. 30, 53–61 (1968)Google Scholar
  48. 48.
    Tsirelson, B.S.: Not every Banach space contains ℓp or c0. Funct. Anal. Appl. 8, 138–141 (1974)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsAristotle University of ThessalonikiGreece

Personalised recommendations