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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Gleason parts and point derivations for uniform algebras with dense invertible group
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by Alexander J. Izzo PDF
Trans. Amer. Math. Soc. 370 (2018), 4299-4321 Request permission

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
  • 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
  • © Copyright 2018 American Mathematical Society
  • 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
  • MathSciNet review: 3811529