The compact Hausdorff space X has the CSWP iff 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 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.