## 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.## References

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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
- © 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

Dedicated: Dedicated to Andrew Browder