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Isolated point theorems for uniform algebras on two- and three-manifolds


Author: Swarup N. Ghosh
Journal: Proc. Amer. Math. Soc. 144 (2016), 3921-3933
MSC (2010): Primary 32E30, 46J10
DOI: https://doi.org/10.1090/proc/13036
Published electronically: March 30, 2016
MathSciNet review: 3513549
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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.


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Swarup N. Ghosh
Affiliation: Department of Mathematics, Southwestern Oklahoma State University, Weatherford, Oklahoma 73096
Email: swarup.ghosh@swosu.edu

DOI: https://doi.org/10.1090/proc/13036
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
Article copyright: © Copyright 2016 American Mathematical Society