The Complex Stone–Weierstrass Property

Kenneth Kunen¹

¹ Department of Mathematics University of Wisconsin Madison, WI 57306, U.S.A.

Fundamenta Mathematicae, Tome 182 (2004) no. 2, pp. 151-167

Résumé

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.

Détail

DOI : 10.4064/fm182-2-5

Keywords: compact hausdorff space has cswp every subalgebra mathbb which separates points contains constant functions dense mathbb results rudin hoffman singer scattered have cswp many non scattered fail cswp whether having cswp just equivalent being scattered here prove general facts about cswp particular compact ordered space has cswp does contain copy cantor set provides class non scattered spaces cswp among these double arrow space aleksandrov urysohn cswp space implies stone weierstrass property complex regulated functions unit interval