On Hoffman's characterization theorem of parts

Abstract

Let $H^\infty(\Delta)$ be the uniform algebra of bounded analytic functions on the open unit disc $\Delta$, and let $\mathfrak{M}(H^\infty)$ be the maximal ideal space of $H^\infty(\Delta)$. Applying Wermer's embedding theorem directly, we investigate the relation between the analytic structure in $\mathfrak{M}(H^\infty)$ and certain separability conditions. Our method rests only on the corona theorem and certain properties of analytic discs. Among other things, without deep factorization theorems on Blaschke products, we derive the famous Hoffman theorem: Let $P(\xi)$ be the Gleason part of $\xi$ in $\mathfrak{M}(H^\infty)$. Then $P(\xi)$ is an analytic disc if and only if $\xi$ lies in the closure in $\mathfrak{M}(H^\infty)$ of an interpolating sequence in $\Delta$.