Dual Algebras and A-Measures

Marek Kosiek,Krzysztof Rudol

doi:10.1155/2014/364271

Marek Kosiek, Krzysztof Rudol

Open Access

https://doi.org/10.1155/2014/364271

Copy DOI

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⊂Cn, 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 onΩand its abstract counterpart—thew* closure of a function algebraAin the dual of the band of measures generated by one of Gleason parts of the spectrum ofA.

Highlights

Let us recall that a uniform algebra over a compact Hausdorff space X is a closed unital subalgebra A of C(X) separating the points of X
We can assume that A is a uniform algebra on its spectrum Sp(A)—the space of nonzero linear and multiplicative homomorphisms of A
Measures on X absolutely continuous with respect to measures representing certain points of G form a band of measures spectrum denoted of some bqyuM otiGen. tThaelgweberaak-osftaAr∗d∗enissiatyn of G in the equivalent formulation of the corona problem in H∞(Ω) settled 50 years ago by Lennart Carleson in the unit disc case and still open for the higher dimensional balls or polydiscs. This alone justifies the need for better understanding of the nature of weak-star closures of nontrivial Gleason parts G canonically embedded in the dual space for MG

Summary

Introduction

Measures on X absolutely continuous with respect to measures representing certain points of G form a band of measures spectrum denoted of some bqyuM otiGen. tThaelgweberaak-osftaAr∗d∗enissiatyn of G in the equivalent formulation of the corona problem in H∞(Ω) settled 50 years ago by Lennart Carleson in the unit disc case and still open for the higher dimensional balls or polydiscs. This alone justifies the need for better understanding of the nature of weak-star closures of nontrivial Gleason parts G canonically embedded in the dual space for MG (or in some other bidual spaces).

Preliminaries

Second Duals

Closures of Parts and Representing Measures

A-Measures