The Complex Stone–Weierstrass Property
Volume 182 / 2004
Abstract
The compact Hausdorff space $X$ has the CSWP iff every subalgebra of $C(X, {\mathbb C})$ which separates points and contains the constant functions is dense in $C(X, {\mathbb C})$. Results of W. Rudin (1956) and Hoffman 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 iff $X$ 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.
