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 Q _I 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.