Characters on singly generated $C^{\ast }$-algebras

In this note we consider the question of what elements $\delta$ in the spectrum of a bounded operator $A$ on Hilbert space have the property that there is a multiplicative linear functional $\phi$ on the ${C^{\ast }}$-algebra generated by $A$ and $I$ whose value at $A$ is $\delta$. If $A$ is hyponormal then there is a character $\phi$ on the ${C^{\ast }}$-algebra generated by $A$ and $I$ such that $\phi (A) = \delta$ if and only if $\delta$ is in the approximate point spectrum of $A$. We use this to prove a structure theorem for the ${C^{\ast }}$-algebra generated by a hyponormal operator. We conclude by proving that any pure state on a Type I ${C^{\ast }}$-algebra is multiplicative on some maximal abelian ${C^{\ast }}$-subalgebra.