CLOSED IDEALS AND THE BENNETT-GILBERT CONJECTURE IN BANACH-ALGEBRAS OF ANALYTIC-FUNCTIONS

Let B be an isometric function algebra on the unit circle T and let B = B boolean AND A ((D) over bar) be the corresponding algebra of anal ytic functions (where A((D) over bar) is the disc algebra). For a clos ed ideal I in B+, let h(I) = {z is an element of (D) over bar : f(z) = 0 for every f is an element of 1} be the hull of I and let Q(I) be th e greatest common divisor of the inner parts of the non-zero functions in I. Moreover, denote by I-B the closed ideal in B generated by I. W e confirm the Bennett-Gilbert conjecture I = Q(I)A((D) over bar) boole an AND I-B under the assumption that h(I) boolean AND T is contained i n the Canter set. This generalizes work of Esterle, Strouse and Zouaki a.