The Complex Stone-Weierstrass Property
by
Kenneth Kunen
University of Wisconsin

Let X be a compact Hausdorff space. By the standard Stone-Weierstrass Theorem, if A is any algebra of continuous real-valued functions on X which separates points and contains the constant functions, then A contains all real-valued functions on X. Now, we say that X has the CSWP iff the same theorem holds for algebras of complex-valued functions on X. By classical results, the CSWP is false of the unit disc in the complex plane. W. Rudin showed that the compact metric space X has the CSWP iff X is scattered.

Here, we prove some general facts about the CSWP; and in particular we show that if X is a compact separable LOTS, then X has the CSWP iff X does not contain a copy of the Cantor set. This provides a class of non-scattered spaces, such as the double arrow space, which have the CSWP.

Date received: May 15, 2003