The Complex Stone{weierstrass Property
نویسنده
چکیده
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; and 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.
منابع مشابه
Complex Function Algebras and Removable Spaces
The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X ...
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The compact Hausdorff space X has the CSWP if every subalgebra of C(X, C) which separates points and contains the constant functions is dense in C(X, C). W. Rudin showed that all scattered X have the CSWP. We describe a class of non-scattered X with the CSWP; by another result of Rudin, such X cannot be metrizable.
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