Dual algebras and A-measures

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. © 2014 marek kosiek and krzysztof rudol.