Dual Algebras and A-Measures. (13th October 2014) - Kosiek, Marek; Rudol, Krzysztof
Abstract : 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 holomorphyΩ ⊂ C n, our approach avoids the use of integral formulae for analytic functions, strict pseudoconvexity, or some other regularity ofΩ . We also investigate the relation between the algebra of bounded holomorphic functions onΩ and its abstract counterpart—thew * 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 .