Topology of Gleason Parts in Maximal Ideal Spaces wit h no Analytic Discs

摘要

We strengthen, in various directions, the theorem of Garnett that every sigma-compact, completely regular space X occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space X is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace X of a Euclidean space, there is a compact set K in some C-N so that (K) over cap K contains a Gleason part homeomorphic to X, and (K) over cap contains no analytic discs.