Convexity of the Spectrum of a Multiplication Operator

Abstract

Let $F$ be a compact subset  of the complex plane, $m$ be the lebesgue measure and $\nu=m|_F$. If $A$ is the multiplication operator on $L^2(\nu)$ and $C^*(A)$ is the $C^*$-algebra generated by $A$, then $F$ is convex if and only if the pure state space of $C^*(A)$ is convex.