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Gleason parts and point derivations for uniform algebras with dense invertible group


Author: Alexander J. Izzo
Journal: Trans. Amer. Math. Soc. 370 (2018), 4299-4321
MSC (2010): Primary 46J10, 46J15, 32E20, 32A65, 30H50
DOI: https://doi.org/10.1090/tran/7153
Published electronically: February 26, 2018
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Abstract: It is shown $ \vphantom {\widehat {\widehat {\widehat {\widehat {\widehat {\widehat {\widehat X}}}}}}}$ that there exists a compact set $ X$ in $ \mathbb{C}^N$ ($ N\geq 2$) such that $ \widehat X\setminus X$ is nonempty and the uniform algebra $ 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 $ X$ such that $ 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 $ \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.


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Additional Information

Alexander J. Izzo
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
Email: aizzo@bgsu.edu

DOI: https://doi.org/10.1090/tran/7153
Keywords: Polynomial convexity, polynomially convex hulls, rational convexity, rationally convex hulls, hull without analytic structure, Gleason parts, point derivations, dense invertibles, antisymmetry, Swiss cheese
Received by editor(s): June 19, 2016
Received by editor(s) in revised form: November 23, 2016
Published electronically: February 26, 2018
Dedicated: Dedicated to Andrew Browder
Article copyright: © Copyright 2018 American Mathematical Society

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