J. Aust. Math. Soc.
75 (2003), 125143

Extreme point methods and BanachStone theorems

Hasan AlHalees
Department of Mathematics
Saginaw Valley State University
Saginaw MI 48710
USA
hhalees@svsu.edu



Richard J. Fleming
Department of Mathematics
Central Michigan University
Mt. Pleasant MI 48859
USA
flemi1rj@cmich.edu



Abstract

An operator is said to be nice if its
conjugate maps extreme points of the dual unit
ball to extreme points. The classical
BanachStone 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 vectorvalued version of the
Choquet Boundary. We also characterize nice
operators from one function module to another.

Download the article in PDF format (size 171 Kb)


