Journal of the Australian Mathematical Society

Table of Contents - 2009 - Volume 87, Issue 02  

Research Article

FLAT SETS, ℓp-GENERATING AND FIXING c0 IN THE NONSEPARABLE SETTING

M. FABIANa1 c1, A. GONZÁLEZa2 and V. ZIZLERa3

a1 Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic (email: fabian@math.cas.cz)

a2 Instituto de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, C/Vera, s/n. 46022 Valencia, Spain (email: algoncor@doctor.upv.es)

a3 Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic (email: zizler@math.cas.cz)

Abstract

We define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak* Kadec–Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c0(ω1)-generated or ℓp(ω1)-generated, where pxs2208(1,∞). In particular, we show that every subspace of c0(ω1) is c0(ω1)-generated and every subspace of ℓp(ω1) is ℓp(ω1)-generated for every pxs2208(1,∞). As a byproduct of the technology of projectional resolutions of the identity we get an alternative proof of Rosenthal’s theorem on fixing c0(ω1).

(Received February 11 2008)

(Accepted November 14 2008)

2000 Mathematics subject classification

  • primary 46B26; secondary 46B03;
  • 46B20

Keywords and phrases

  • Lipschitz-weak*-Kadets–Klee norm;
  • c0(Γ)-generated space;
  • ℓp(Γ)-generated space;
  • weakly compactly generated space;
  • asymptotically p-flat set;
  • innerly asymptotically p-flat set

Correspondence:

c1 For correspondence; e-mail: fabian@math.cas.cz

Footnotes

The first author was supported by grants AVOZ 101 905 03 and IAA 100 190 610 and the Universidad Politécnica de Valencia. The second author was supported in a Grant CONACYT of the Mexican Government. The third author was supported by grants AVOZ 101 905 03 and GAČR 201/07/0394.