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

We characterize the algebra $H^{\mathrm{\infty }}\circ L_m$, where $m$ is a point of the maximal ideal space of $H^{\mathrm{\infty }}$ 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 $f\circ L_{m} \in H^\infty$thereexists$F\in H^\infty$suchthat$F|_{P(m)} = f$.