  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 $\scr F$ of perfect subsets of $\Gamma$, including the
triadic Cantor set, such that the above equality holds whenever
$h(I)\cap\Gamma\in\scr F$. We also give counterexamples to the conjecture.
