The Complex Stone–Weierstrass Property

Abstract

The compact Hausdor space X has the CSWP i every subalgebra of C(X;C) which separates points and contains the constant functions is dense in C(X;C). Results of W. Rudin (1956) and Homan and Singer (1960) show that all scattered X have the CSWP and many non-scattered X fail the CSWP, but it was left open whether having the CSWP is just equivalent to being scattered. Here, we prove some general facts about the CSWP; in particular we show that if X is a compact ordered space, then X has the CSWP i X does not contain a copy of the Cantor set. This provides a class of non-scattered spaces with the CSWP. Among these is the double arrow space of Aleksandrov and Urysohn. The CSWP for this space implies a Stone{Weierstrass property for the complex regulated functions on the unit interval. 1. Introduction. All spaces discussed in this paper are Hausdor. Definition 1.1. If X is compact, then C(X) = C(X;C) is the algebra of continuous complex-valued functions on X, with the usual supremum norm.Av C(X) means thatA is a subalgebra of C(X) which separates points and contains the constant functions. IfAv C(X) is self-adjoint (f 2A$ f 2A), thenA is dense in C(X) by the standard Stone{Weierstrass Theorem for real-valued functions. Definition 1.2. The compact space X has the Complex Stone{Weier- strass Property (CSWP) i everyAv C(X) is dense in C(X). The CSWP may be considered a notion of \smallness by the following easy