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Spaces of Continuous Vector Functions as Duals

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 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

46E4046E1546G10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 1 , 01 March 1988 , pp. 70 - 78
Copyright
Copyright © Canadian Mathematical Society 1988

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