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Strong Regularity at Nonpeak Points
Abstract
We construct a uniform algebra which is strongly regular at a nonpeak point.
... In [25], Wang uses McKissick's lemma to give an example of a Swiss cheese set X such that R(X) is strongly regular at a non-peak point. In [9], the first author gave an example of a Swiss cheese set X such that R(X) has no non-trivial Jensen measures yet is not regular. ...
Swiss cheese sets have been used in the literature as useful examples in the
study of rational approximation and uniform algebras. In this paper, we give a
survey of Swiss cheese constructions and related results. We describe some
notable examples of Swiss cheese sets in the literature. We explain the various
abstract notions of Swiss cheeses, and how they can be manipulated to obtain
desirable properties. In particular, we discuss the Feinstein-Heath
classicalisation theorem and related results. We conclude with the construction
of a new counterexample to a conjecture of S. E. Morris, using a classical
Swiss cheese set.
... Wang gives an example in [9] to show that (v) does not imply (iii). But whether or not either of the implications (v) ⇒ (iv) or (iv) ⇒ (iii) is true for uniform algebras appears to be open. ...
There are many forms of regularity conditions which a Banach function algebra (i.e. a commutative, semisimple Banach algebra) may satisfy. Many of these conditions have their origins in harmonic analysis, and now have applications in the theory of automatic continuity, and the theory of Wedderburn (and strong Wedderburn) decomposability of commutative Banach algebras. We shall discuss the relationships between these conditions in the general setting and also for uniform algebras. We shall include in our discussion strong regularity, Ditkin’s condition, bounded relative units, and maximal ideals which have a bounded approximate identity.
... There are also versions of (1) and (2) asked at one point at a time. In [5], Wang gave an example of a compact set X C C with 0 G X such that R(X) is strongly regular at 0 but M o does not have a bounded approximate identity. (Here R[X) is the uniform algebra of all / G C(X) which may be uniformly approximated by rational functions with no poles in X). ...
In this note we answer a question of W. G. Bade by showing that if a normal, unital Banach function algebra A is strongly regular at one of its characters φ has a bounded approximate identity, then A has bounded relative units at φ. In particular, every strong Ditkin algebra has bounded relative units at all points of its character space. There need not, however, be a global bound available for the relative units.
In this paper we answer a question of Wilken by constructing an example of a non-trivial, strongly regular uniform algebra on a compact, Hausdorff; topological space. We also show how to obtain an example on a compact metric space, and we give examples both with and without the property that every point is a peak point.
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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