Operating functions and subspaces of C_0(X)
by
Eggert Briem
Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland

A continuous function h defined on a neigbourhood of 0 in the real line is said to operate on a subspace B of C₀(X), the space of all continuous real-valued functions on a locally compact Hausdorff space X, if the composite function h o b is in B whenever b is in B and the composition is defined.

If X is compact and B contains the constant functions then, unless the uniform closure of B contains every continuous function on X which identifies points identified by B, there are only trivial operating functions for B, i.e. functions of the form h(t)=\alphat+\beta.

In the locally compact case, where the constant functions are not in C₀(X), the situation is different, it could f. ex. happen that b(x₁)=\lambdab(x₂) for all b in B in which case any operating function h for B must satisfy h(\lambdat)=\lambdah(t) for t in a neighbourhood of 0, in particular h must be odd if \lambda = -1.

One such function for positive \lambda is the function h(t)=|t|. If this function operates on B, in other words if B is a lattice, every f in C₀(X) which can be approximated on every pair of points in X can be uniformly approximated by functions in B. We show among other things that this statement remains true with h(t)=|t| replaced by a function h satisfying |h(t)| <= k|t| for t in a neighbourhood of 0, a condition which is f. ex. satisfied if h(\lambdat)=\lambdah(t) in a neighbourhood of 0.

(T)

Date received: February 2, 2000

Copyright © 2000 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 # caei-06.