Additive isometries of C(X)
by
Lawrence Narici
St. John's University

For compact Hausdorff spaces X and Y, the Stone-Banach theorem asserts that linear isometries H of C(X) onto C(Y) are of the form Hf(y) = f(h(y)) H1(y) (f in C(X), y in Y) where h : Y --> X is a homeomorphism and | H1(y) | \equiv 1. Maps f --> (f o h) H1 of this type, are called `weighted composition maps' where H1 in C(Y) is the `weight' function. If, instead of R- or C-valued continuous functions, we take C(X) and C(Y) to consist of K-valued continuous functions where (K, | ·|) is a valued field, linear isometries may take different forms. Indeed, if (K, | ·|) is non-Archimedean, a linear isometry H : C(X) --> C(Y) is a weighted composition if and only if it is separating in the sense that, for all f, g in C(X), fg = 0 ===> HfHg = 0. We weaken linear to additive and isometry to bijection and consider what forms additive separating bijections H have in this article for K = R, C or a non-Archimedean valued field. We show that an additive separating bijection H : C(X) --> C(Y) is automatically continuous and must `almost' be a weighted composition map with a homeomorphism. The form that H takes depends on the field K in which the functions take values. The sharpest statements can be made when H is an isometry.

Date received: February 9, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caak-20.