Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-23T19:27:19.475Z Has data issue: false hasContentIssue false
  • English
  • Français

ON λ-STRICT IDEALS IN BANACH SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  27 September 2010

TROND A. ABRAHAMSEN  and
OLAV NYGAARD
TROND A. ABRAHAMSEN*
Affiliation:
Department of Mathematics, University of Agder, Postbox 422, 4604 Kristiansand, Norway (email: Trond.A.Abrahamsen@uia.no)
OLAV NYGAARD
Affiliation:
Department of Mathematics, University of Agder, Postbox 422, 4604 Kristiansand, Norway (email: Olav.Nygaard@uia.no)
*
✉For correspondence; e-mail: trond.a.abrahamsen@uia.no
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

Keywords

strict idealu-idealVN-subspace

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 231 - 240
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Abrahamsen, T. A., Lima, V. and Lima, Å., ‘Unconditional ideals of finite rank operators’, Czechoslovak Math. J. 58 (2008), 1257–1278.CrossRefGoogle Scholar
[2]Bandyopadhyay, P., Basu, S., Dutta, S. and Lin, B.-L., ‘Very non-constrained subspaces of Banach spaces’, Extracta Math. 18(2) (2003), 161–185.Google Scholar
[3]Belobrov, P. K., ‘Minimal extension of linear functionals to second dual spaces’, Mat. Zametki 27(3) (1980), 439–445 (in Russian); English translation in: Math. Notes 27(3–4) (1980), 218–221.Google Scholar
[4]Godefroy, G. and Kalton, N. J., ‘Approximating sequences and bidual projections’, Q. J. Math. Oxford Series (2) 48(190) (1997), 179–202.CrossRefGoogle Scholar
[5]Godefroy, G., Kalton, N. J. and Saphar, P. D., ‘Unconditional ideals in Banach spaces’, Studia Math. 104 (1993), 13–59.CrossRefGoogle Scholar
[6]Godefroy, G. and Saphar, P., ‘Duality in spaces of operators and smooth norms on Banach spaces’, Illinois J. Math. 32 (1988), 672–695.Google Scholar
[7]James, R. C., ‘A non-reflexive Banach space isometric with its second conjugate space’, Proc. Natl. Acad. Sci. USA 37 (1951), 174–177.Google Scholar
[8]James, R. C., ‘A separable somewhat reflexive space with non-separable dual’, Bull. Amer. Math. Soc. 80 (1974), 738–743.Google Scholar
[9]Lima, V. and Lima, Å., ‘A three-ball intersection property for u-ideals’, J. Funct. Anal. 252(1) (2007), 220–232.CrossRefGoogle Scholar
[10]Lima, V. and Lima, Å., ‘Strict u-ideals in Banach spaces’, Studia Math. 195(3) (2009), 275–285.Google Scholar
[11]Maurey, B., ‘Types and ℓ 1-subspaces’, in: Texas Functional Analysis Seminar, Longhorn Notes (eds. Odell, E. and Rosenthal, H. P.) (The University of Texas at Austin, 1982–1983), pp. 123–137.Google Scholar
[12]Nygaard, O., ‘Thick sets in Banach spaces and their properties’, Quaest. Math. 29 (2006), 59–72.CrossRefGoogle Scholar