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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
MathSciNet review: 3811529
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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
MR Author ID: 307587
Email: aizzo@bgsu.edu

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