On spectral theory and convexity

Abstract:

A compact convex set $K$ in a locally convex algebra is said to be a spectral carrier if, for all $x$, $y \in K$, we have $xy = yx \in K$ and $x + y - xy \in K$. We show that if a compact convex set $K$ is a spectral carrier, then the idempotents in $K$ are exactly the extreme points of $K$ and form a complete lattice. Conversely, if a compact set $K$ is a closed convex hull of a lattice of commuting idempotents, then $K$ is a spectral carrier. Furthermore, a metrizable spectral carrier is a Choquet simplex if and only if its extreme points form a chain of idempotents.