Hostname: page-component-68945f75b7-qf55q Total loading time: 0 Render date: 2024-08-05T14:27:37.689Z Has data issue: false hasContentIssue false
  • English
  • Français

On the weak*-Radon Nikodym property

Published online by Cambridge University Press:  17 April 2009

Elias Saab
Elias Saab
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211
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 say that a certain property in a Banach space E is stable by subspaces if every closed subspace of E enjoys the same property. It is well known that the Radon-Nikodym property is stable by subspaces while the Weak Radon-Nikodym property is not. In his recent memoir, Talagrand investigated the stability of the Weak*Radon-Nikodym property which is a generalization of the Weak Radon-Nikodym property and showed that under Axiom L, the Weak*Radon-Nikodym property is stable by subspaces. It is still an open problem whether or not this result holds without this extra set theoretical hypothesis. In this paper we show that in a dual Banach space, the Weak*Radon-Nikodym property is stable by subspaces without assuming Axiom L. Other related results are discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 3 , June 1988 , pp. 323 - 332
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Diestel, J. and Uhl, J.J. Jr., Vector measures, Math Surveys 15 (Amer. Math. Soc., Providence, RI, 1977).CrossRefGoogle Scholar
[2]Fremlin, D. and Talagrand, M., ‘A decomposition theorem for additive set functions and applications to Pettis integrals and erogodic means’, Math. Z. 168 (1979), 117–142.CrossRefGoogle Scholar
[3]Ghoussoub, N. and Saab, E., ‘On the weak Radon Nikodym property’, Proc. Amer. Math. Soc. 81 (1981), 81–84.Google Scholar
[4]Riddle, L. and Saab, E., ‘On functions that are universally Pettis integrable’, Illinois J. Math. 29 (1985), 509–531.CrossRefGoogle Scholar
[5]Rosenthal, H., ‘A characterization of Banach spaces containing ℓ1‘, Proc. Nat. Acad. Sci., U.S.A. 71 (1974), 2411–2413.CrossRefGoogle Scholar
[6]Saab, E., ‘Some characterizations of weak Radon Nikodyni sets’, Proc. Amer. Math. Soc. 86 (1982), 307–311.Google Scholar
[7]Schachermayer, W., ‘Operators from L 1 to Banach spaces and subsets of L ∞’, (pre-print).Google Scholar
[8]Talagrand, M., Pettis integral and measure theory 51, Memoirs of the Amer. Math Soc., 1984.Google Scholar