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Extreme Point Methods and Banach-Stone Theorems

Part of: Linear function spaces and their duals Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

Hasan Al-Halees  and
Richard J. Fleming
Hasan Al-Halees
Affiliation:
Department of Mathematics Saginaw Valley State UniversitySaginaw MI 48710USA e-mail: hhalees@svsu.edu
Richard J. Fleming
Affiliation:
Department of Mathematics Central Michigan UniversityMt. Pleasant MI 48859USA e-mail: flemilrj@cmich.edu
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Abstract

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An operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of C0( Q, X) into C0(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.

MSC classification

Secondary: 46B04: Isometric theory of Banach spaces 46E40: Spaces of vector- and operator-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 1 , August 2003 , pp. 125 - 143
Copyright
Copyright © Australian Mathematical Society 2003

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