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A note on a construction of J. F. Feinstein

M. J. Heath — 2005

Studia Mathematica

In [6] J. F. Feinstein constructed a compact plane set X such that R(X), the uniform closure of the algebra of rational functions with poles off X, has no non-zero, bounded point derivations but is not weakly amenable. In the same paper he gave an example of a separable uniform algebra A such that every point in the character space of A is a peak point but A is not weakly amenable. We show that it is possible to modify the construction in order to produce examples which are also regular.

Swiss cheeses, rational approximation and universal plane curves

J. F. Feinstein;  M. J. Heath — 2010

Studia Mathematica

We consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We develop a theory of allocation maps connected to such sets and we use this theory to modify examples previously constructed in the literature to obtain examples homeomorphic to the Sierpiński carpet. Our techniques also allow us to avoid certain technical difficulties in the literature.

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