Closed ideals in the bidise algebra

Let D be the open unit disc in C, T=0D the unit circle, and let D"=DX... • be the n-dimensional unit polydisc. Let H ~ (D") be the algebra of bounded analytic functions on D", endowed with the uniform norm on D". The polydisc algebra is the space A(D")=C(D")nH~'(D"), also given the uniform norm on D". The spaces A(D) and A(D 2) are known as the disc and bidisc algebras, respectively. Let us introduce a weak-star topology on H = (D"). The space L ~ * (T") is the dual space of LI(T"), so it has a weak-star topology. One can think of H*~(D ") as a sub-space of L = (T") via radial limits, and as such it is weak-star closed. We define the weak-star topology on H=(D ") by saying that a set UcH~(D ") is open if there is a weak-star open set VcL~(T ") with U = VnH ~ (D"). For a collection ~ " of functions in A(D"), associate the zero set Z( ~ r) = {zED": f(z) = 0 for all fE~}, and if Ec~ " , introduce the closed ideal J (e) = {fEa(Dn): f = 0 on E}. In this paper, we will try to describe the closed ideals of the bidisc algebra. The result we obtain is the following. Every closed ideal I in the bidisc algebra A (D ~) has the form I = J(Z(I) n T') ('I{fEA(Da): f(a,-)Eu, H*O(D) and f(.,a)Ev, H*O(D) for all aET} n [I]w,, where u, and v, are inner functions in H**(D) for each ~ET, and [I]w * is the weak-star closure of I in H~ * (D~). It is easy to see that [I]w * is a (weak-star closed) idea