Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T19:39:43.006Z Has data issue: false hasContentIssue false

Spaces of Continuous Vector Functions as Duals

Published online by Cambridge University Press:  20 November 2018

Michael Cambern
Affiliation:
Department of Mathematics, University of California Santa Barbara, CA 93106
Peter Greim
Affiliation:
Department of Mathematics The Citadel Charleston, SC 29409
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.

A well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Behrends, E., et al., L -structure in real Banach spaces, Lecture Notes in Mathematics 613, Springer-Verlag, Berlin-Heidelberg-New York, 1977.Google Scholar
2. Cambern, M. and Greim, P., The bidual of C(X, E), Proc. Amer. Math. Soc. 85 (1982), pp. 5358.Google Scholar
3. Cambern, M. and Greim, P., The dual of a space of vector measures, Math. Z. 180 (1982), pp. 373378.Google Scholar
4. Cembranos, P., C(K, E) contains a complemented copy of c§, Proc. Amer. Math. Soc. 91 (1984), pp. 556558.Google Scholar
5. Day, M. M., Normed linear spaces, 3rd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1973.Google Scholar
6. Diestel, J. and Uhl, J. J. Jr., Vector measures, Math. Surveys 15, Amer. Math. Soc, Providence, R.I., 1977.Google Scholar
7. Dinculeanu, N., Vector measures, Pergamon Press, New York, 1967.Google Scholar
8. Dixmier, J., Sur certains espaces considérés par Stone M. H., Summa Brasil. Math. 2 (1951), pp. 151182.Google Scholar
9. Dunford, N. and Schwartz, J. T., Linear operators, Part I, Interscience, New York, 1958.Google Scholar
10. Greim, P., Banach spaces with the Lx∼ Banach-S tone property, Trans. Amer. Math. Soc. 287 (1985), pp. 819828.Google Scholar
11. Grothendieck, A., Une caractérisation vectorielle métrique des espaces L1 , Canadian J. Math. 7 (1955), pp. 552561.Google Scholar
12. Holmes, R. B., Geometric functional analysis and its applications, Springer-Verlag, Berlin- Heidelberg-New York, 1975.Google Scholar
13. Lacey, H. E., The isometrical theory of classical Banach spaces, Springer-Verlag, Berlin- Heidelberg-New York, 1974.Google Scholar
14. Singer, I., Linear functional on the space of continuous mappings of a compact space into a Banach space, Rev. Roumaine Math. Pures Appl. 2 (1957), pp. 301315. (Russian)Google Scholar
15. Taylor, A. E., Introduction to functional analysis, John Wiley and Sons, New York, 1958.Google Scholar