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Homeomorphic Analytic Maps into the Maximal Ideal Space of H∞

Published online by Cambridge University Press:  20 November 2018

Daniel Suárez
Daniel Suárez*
Affiliation:
Departamento de Matemática, Fac. de Cs. Exactas y Naturales, UBA, Pab. I, Ciudad Universitaria, (1428) Núñez, Capital Federal, Argentina email: [email protected]
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Abstract

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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 $f\,\in \,{{H}^{\infty }}\,\text{to}\,{{I}_{m}}$.

Keywords

30H0546J20
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 1 , 01 February 1999 , pp. 147 - 163
Copyright
Copyright © Canadian Mathematical Society 1999

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