Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T00:56:29.236Z Has data issue: false hasContentIssue false
  • English
  • Français

Isometric Preduals of James Spaces

Published online by Cambridge University Press:  20 November 2018

Leon Brown  and
Takashi Ito
Leon Brown
Affiliation:
Wayne State University, Detroit, Michigan
Takashi Ito
Affiliation:
Wayne State University, Detroit, Michigan
Rights & Permissions [Opens in a new window]

Extract

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.

A Banach space B is called an isometric predual, or simply a predual, of a Banach space X if the dual B* of B is isometrically isomorphic to X. A Banach space X is said to have a unique (isometric) predual if X has a predual and all preduals are mutually isometrically isomorphic. In general a Banach space does not have a unique predual even if it has a predual. A simple example of this is the space l1, because c0 and c are isometric preduals of l1 but not isometrically isomorphic. A. Grothendieck [3] first noticed that L∞-spaces have unique preduals, and then S. Sakai generalized this to von Neumann algebras (see p. 30 of [9]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 59 - 69
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Ando, T., Uniqueness of predual of Hx (to appear).Google Scholar
2. Day, M. M., Normed linear spaces (third éd., New York, 1973).Google Scholar
3. Grothendieck, A., Une caracterisation vectorielle-métrique de espaces L1, Can. J. Math. 7 (1955), 552–561.Google Scholar
4. Ito, T., On Banach spaces with unique isometric preduals, Mich. Math. J. 24 (1977), 321–324.Google Scholar
5. James, R. C., Bases and reflexivity of Banach spaces, Ann. of Math. 52 (1950), 518–527.Google Scholar
6. James, R. C., A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174–177.Google Scholar
7. James, R. C., Banach spaces quasi-reflexive of order one, Studia Math. 60 (1977), 157–177.Google Scholar
8. Lindenstrauss, J., On complemented subspace of m, Israel J. Math. 5 (1967), 153–156.Google Scholar
9. Sakai, S., C*-algebras and W*-algebras (Springer-Verlag, Berlin, 1971).Google Scholar