Strong regularity for uniform algebras

Details Strong regularity for uniform algebras Strong regularity for uniform algebras

1998-09-14

arxiv.org/abs/math/9809073v1

Mathematics

Functional Analysis

AMSTEX, 11 pages

Scientific paper

A survey is given of the work on strong regularity for uniform algebras over the last thirty years, and some new results are proved, including the following. Let A be a uniform algebra on a compact space X and let E be the set of all those points x of X such that A is not strongly regular at x. If E has no non-empty, perfect subsets then A is normal, and X is the character space of A. If X is either the interval or the circle and E is meagre with no non-empty, perfect subsets then A is trivial. These results extend Wilken's work from 1969. It is also shown that every separable Banach function algebra which has character space equal to either the interval or the circle and which has a countably-generated ideal lattice is uniformly dense in the algebra of all continuous functions.

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