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

Ghosh, Swarup

doi:10.1090/proc/13036

Isolated point theorems for uniform algebras on two- and three-manifolds
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Proc. Amer. Math. Soc. 144 (2016), 3921-3933 Request permission

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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