Closed ideals of the algebra of absolutely convergent Taylor series

Abstract:

Let $\Gamma$ be the unit circle, $A(\Gamma )$ the Wiener algebra of continuous functions whose series of Fourier coefficients are absolutely convergent, and ${A^ + }$ the subalgebra of $A(\Gamma )$ of functions whose negative coefficients are zero. If I is a closed ideal of ${A^ + }$, we denote by ${S_I}$ the greatest common divisor of the inner factors of the nonzero elements of I and by ${I^A}$ the closed ideal generated by I in $A(\Gamma )$. It was conjectured that the equality ${I^A} = {S_I}{H^{\infty }} \cap {I^A}$ holds for every closed ideal I. We exhibit a large class ${\mathcal {F}}$ of perfect subsets of $\Gamma$, including the triadic Cantor set, such that the above equality holds whenever $h(I) \cap \Gamma \in {\mathcal {F}}$. We also give counterexamples to the conjecture.