Gleason parts and COP

Abstract

Let H ∞ denote the Banach algebra of bounded analytic functions on the open unit disc D , and let M( H ∞) denote the maximal ideal space of H ∞. One of the results in this paper shows that if two Gleason parts (in M( H ∞)) have distinct closures, then there exists an H ∞ function which is identically zero on one part and nonzero on the other. From this one obtains a Corona theorem for certain subalgebras of H ∞. Let COP denote the algebra of bounded analytic functions that are in the little Bloch space. It is shown here that there exists a COP function h and an inner function I such that I divides h in H 1, but the result is not in COP.