On the analytic structure of the $H^{\infty }$ maximal ideal space

Abstract

We characterize the algebra $H^{\infty }\circ L_m$, where $m$ is a point of the maximal ideal space of $H^{\infty }$ with nontrivial Gleason part $P(m)$ and $L_m:\mathbb{D}\to P(m)$ is the coordinate Hoffman map. In particular, it is shown that for any continuous function $f:P(m)\to \mathbb{C}$ with $f\circ L_m\in H^{\infty }$ there exists $F\in H^{\infty }$ such that $F{|}_{P(m)}=f$.

18 pages