The Complex Stone–Weierstrass Property

Tom 182 / 2004

Kenneth Kunen Fundamenta Mathematicae 182 (2004), 151-167 MSC: Primary 46J10, 54C35; Secondary 46E25. DOI: 10.4064/fm182-2-5


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.


  • Kenneth KunenDepartment of Mathematics
    University of Wisconsin
    Madison, WI 57306, U.S.A.

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