Isolated point theorems for uniform algebras on two- and three-manifolds
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Abstract:
In 1957, Andrew Gleason conjectured that if $A$ is a uniform algebra on its maximal ideal space $X$ and every point of $X$ is a one-point Gleason part for $A$, then $A$ must contain all continuous functions on $X$. However, in 1968, Brian Cole produced a counterexample to disprove Gleason’s conjecture. In this paper, we establish that Gleason’s conjecture still holds for two important classes of uniform algebras considered by John Anderson, Alexander Izzo and John Wermer in connection with the peak point conjecture. In fact, we prove stronger results by weakening the hypothesis of Gleason’s conjecture for those two classes of uniform algebras.References
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Additional Information
- Swarup N. Ghosh
- Affiliation: Department of Mathematics, Southwestern Oklahoma State University, Weatherford, Oklahoma 73096
- Email: swarup.ghosh@swosu.edu
- Received by editor(s): June 14, 2015
- Received by editor(s) in revised form: June 22, 2015, and November 12, 2015
- Published electronically: March 30, 2016
- Communicated by: Pamela B. Gorkin
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3921-3933
- MSC (2010): Primary 32E30, 46J10
- DOI: https://doi.org/10.1090/proc/13036
- MathSciNet review: 3513549