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Gleason parts and point derivations for uniform algebras with dense invertible group
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Gleason parts and point derivations for uniform algebras with dense invertible group
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Journal Article
Gleason parts and point derivations for uniform algebras with dense invertible group
2018
Overview
It is shownX^^^\\vphantom {\\widehat {\\widehat {\\widehat {\\widehat {\\widehat {\\widehat {\\widehat X}}} that there exists a compact set XX in CN\\mathbb {C}^N (N≥2N\\geq 2) such that X^∖X\\widehat X\\setminus X is nonempty and the uniform algebra P(X)P(X) has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese XX such that R(X)R(X) has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in CN\\mathbb {C}^N is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein.
Publisher
American Mathematical Society
Subject
