Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

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 $X$. Gleason’s conjecture was disproved by Brian Cole in 1968. In this paper, we establish a strengthened form of Gleason’s conjecture for uniform algebras generated by real-analytic functions on compact subsets of real-analytic three-dimensional manifolds-with-boundary.