Maximal weak-$\sp{\ast} $ Dirichlet algebras

Abstract

The purpose of this note is to demonstrate, in the context of weak-* Dirichlet algebras, the interdependence of a number of properties possessed by the space of bounded analytic functions in the open Linit disc. The space of bounded analytic functions in the open unit disc H' has a number of properties, among which are these: (i) H' is an integral domain; (ii) the boundary values of any nonzero function in H' cannot vanish on a set of positive Lebesgue measure; and (iii) the space of boundary functions of functions in H'C forms a maximal weak-* closed subaigebra in LX (of Lebesgue measure on the unit circle). Property (i) is quite elementary, while the other two are fairly deep facts and one would not expect much of a relation to hold between them. Surprisingly, however, from an axiomatic point of view, these three properties are equivalent. We shall show this in the context of weak-* Dirichlet algebras which were introduced by Srinivasan and Wang (4). Recall that by definition a weak-* Dirichlet algebra is an algebra X of essentially bounded measurable functions on a probability measure space