Closed ideals in the bidisc algebra

Abstract

Let D be the open unit disc in C, T = 0 D the unit circle, and let D " = D X ... • 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 subspace 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 U c H ~ ( D ") is open i f there is a weak-star open set V c L ~ ( T ") with U= VnH~ (D"). For a collection ~" o f functions in A(D"), associate the zero set