J. Aust. Math. Soc.
75 (2003), 125-143
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Extreme point methods and Banach-Stone theorems
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Hasan Al-Halees
Department of Mathematics
Saginaw Valley State University
Saginaw MI 48710
USA
hhalees@svsu.edu
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Richard J. Fleming
Department of Mathematics
Central Michigan University
Mt. Pleasant MI 48859
USA
flemi1rj@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
of
into
require
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.
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