$D$-domains and the corona

Abstract:

Let D be a bounded domain in the complex plane C. Let ${H^\infty }(D)$ denote the usual Banach algebra of bounded analytic functions on D. The Corona Conjecture asserts that D is $\text {weak}^\ast$ dense in the space $\mathfrak {M}(D)$ of maximal ideals of ${H^\infty }(D)$. In [2] Carleson proved that the unit disk ${\Delta _0}$ is dense in $\mathfrak {M}({\Delta _0})$. In [7] Stout extended Carleson’s result to finitely connected domains. In [4] Gamelin showed that the problem is local. In [1] Behrens reduced the problem to very special types of infinitely connected domains and established the conjecture for a large class of such domains. In this paper we extract some of the crucial ingredients of Behrens’ methods and extend his results to a broader class of infinitely connected domains.