Dual Algebras and A-Measures - Marek Kosiek; Krzysztof Rudol (Journal of Function Spaces)
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⊂Cn, 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.