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Homeomorphic Analytic Maps into the Maximal Ideal Space of H∞
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $m$ be a point of the maximal ideal space of
${{H}^{\infty }}$ with nontrivial Gleason part
$P\left( m \right)$. If
${{L}_{m}}\,:\,\text{D}\,\to \,\text{P(m)}$ is the Hoffman map, we show that
${{H}^{\infty }}\,\circ \,{{L}_{m}}$ is a closed subalgebra of
${{H}^{\infty }}$. We characterize the points
$m$ for which
${{L}_{m}}$ is a homeomorphism in terms of interpolating sequences, and we show that in this case
${{H}^{\infty }}\,\circ \,{{L}_{m}}$ coincides with
${{H}^{\infty }}$. Also, if
${{I}_{m}}$ is the ideal of functions in
${{H}^{\infty }}$ that identically vanish on
$P\left( m \right)$, we estimate the distance of any
$f\,\in \,{{H}^{\infty }}\,\text{to}\,{{I}_{m}}$.
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- Research Article
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- Copyright © Canadian Mathematical Society 1999
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