Homeomorphic Analytic Maps into the Maximal Ideal Space of $H^\infty$

http://dx.doi.org/10.4153/CJM-1999-009-5
Canad. J. Math. 51(1999), 147-163

  Published:1999-02-01
 Printed: Feb 1999

Let $m$ be a point of the maximal ideal space of $\papa$ with nontrivial Gleason part $P(m)$. If $L_m \colon \disc \rr P(m)$ is the Hoffman map, we show that $\papa \circ L_m$ is a closed subalgebra of $\papa$. We characterize the points $m$ for which $L_m$ is a homeomorphism in terms of interpolating sequences, and we show that in this case $\papa \circ L_m$ coincides with $\papa$. Also, if $I_m$ is the ideal of functions in $\papa$ that identically vanish on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$.