J. Aust. Math. Soc.  75 (2003), 125-143
Extreme point methods and Banach-Stone theorems

Hasan Al-Halees
  Department of Mathematics
  Saginaw Valley State University
  Saginaw MI 48710
Richard J. Fleming
  Department of Mathematics
  Central Michigan University
  Mt. Pleasant MI 48859

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 $C_{0}(Q,X)$ into $C_{0}(K,Y)$ 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.
Download the article in PDF format (size 171 Kb)

TeXAdel Scientific Publishing ©  Australian MS