Dual Algebras and A-Measures

Abstract EN

Weak-star closures of Gleason parts in the spectrum of a function algebra are studied.

These closures relate to the bidual algebra and turn out both closed and open subsets of a compact hyperstonean space.

Moreover, weak-star closures of the corresponding bands of measures are reducing.

Among the applications we have a complete solution of an abstract version of the problem, whether the set of nonnegative A-measures (called also Henkin measures) is closed with respect to the absolute continuity.

When applied to the classical case of analytic functions on a domain of holomorphy Ω ⊂ C n , our approach avoids the use of integral formulae for analytic functions, strict pseudoconvexity, or some other regularity of Ω .

We also investigate the relation between the algebra of bounded holomorphic functions on Ω and its abstract counterpart—the w * closure of a function algebra A in the dual of the band of measures generated by one of Gleason parts of the spectrum of A.