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Uniqueness of Preduals for Spaces of Continuous Vector Functions

Published online by Cambridge University Press:  20 November 2018

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

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A. Grothendieck has shown that if the space C(X) is a Banach dual then X is hyperstonean; moreover, the predual of C(X) is strongly unique. In this article we give a vector analogue of Grothendieck's result. We show that if E* is a reflexive Banach 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 the full content of Grothendieck's theorem for C(X) can be established for C(X,(E*,σ*)). This improves a result previously obtained for the case in which E* is Hilbert space.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 1 , 01 March 1989 , pp. 98 - 104
Copyright
Copyright © Canadian Mathematical Society 1989

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