Gleason parts and Choquet boundary of a~function algebra on a~convex compactum

E. L. Arenson

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 204-207

Résumé

Let $K$ be a convex compactum in a complex locally convex space $E$, $P(K)$ be the uniform algebra of functions on $K$ generated by the restrictions of complexaffine continuous functions on $E$. For $x,y\in E$, we set $H(x,y)=\{(1-\lambda)x+\lambda y\colon\lambda\in\mathbb C\}$. It is proved that: (a) the space of maximal ideals of the algebra $P(K)$ coincides with $K$; (b) distinct points $x,y$ from $K$ belong to the same Gleason part if and only if $x$ and $y$ are relatively interior points of the set $H(x,y)\cap K$ (as a subset of $H(x,y)$); (c) the Choquet boundary of the algebra $P(K)$ coincides with the set of complex-extreme points of the compactum $K$ (that is, of points $x$ not belonging to the relative interior of any set of the form $H(x,y)\cap K$ for $y\ne x$).