Closed ideals and the Bennett-Gilbert conjecture in Banach algebras of analytic functions

Abstract

Let \( {\cal B} \) be an isometric function algebra on the unit circle \( {\Bbb T} \) and let \( {\cal B}^+={\cal B}\cap {\cal A}(\overline {\Bbb D})\) be the corresponding algebra of analytic functions (where \( {\cal A}(\overline {\Bbb D}) \) is the disc algebra). For a closed ideal I in \( {\cal B}^+, \) let \( h(I)=\{z\in {\overline {\Bbb D}}:f(z)=0 \) for every \( f \in I\} \) be the hull of I and let QI be the greatest common divisor of the inner parts of the non-zero functions in I. Moreover, denote by \( I^{\cal B} \) the closed ideal in \( {\cal B} \) generated by I. We confirm the Bennett-Gilbert conjecture¶¶\( I=Q_I{{\cal A}}(\overline {\Bbb D})\cap I^{\cal B} \)¶¶under the assumption that \( h(I)\cap {\Bbb T} \) is contained in the Cantor set. This generalizes work of Esterle, Strouse and Zouakia.