Ideals in big Lipschitz algebras of analytic functions

For 0 < gamma less than or equal to 1, let Lambda(gamma)(+) be the big Lipschitz algebra of functions analytic on the open unit disc D which satisfy a Lipschitz condition of order gamma on (D) over bar. For a closed set E on the unit circle T and an inner function Q, let Jgamma (E, Q) be the closed ideal in Lambda(gamma)(+) consisting of those functions f is an element of Lambda(gamma)(+) for which (i) f = 0 on E, (ii) \f (z) - f (w)\ = o(\z - w\(gamma)) as d(z, E), d(w, E) --> 0, (iii) f/Q is an element of Lambda(gamma)(+) Also, for a closed ideal I in Lambda(gamma)(+), let E-I = {z is an element of T : f(z) = 0 for every f is an element of I} and let Q(1) be the greatest common divisor of the inner parts of non-zero functions in I. Our main conjecture about the ideal structure in Lambda(gamma)(+) is that J(gamma)(E-I, Q(I)) subset of or equal to I for every closed ideal I in Lambda(gamma)(+). We confirm the conjecture for closed ideals I in Lambda(gamma)(+) for which E-I is countable and obtain partial results in the case where Q(I) = 1. Moreover, we show that every wk* closed ideal in Lambda(gamma)(+) is of the form {f is an element of Lambda(gamma)(+) : f = 0 on E and f/Q is an element of Lambda(gamma)(+)} for some closed set E subset of or equal to T and some inner function Q.