Open Access
2015 Absolute continuity of positive linear functionals
Zsolt Szűcs
Banach J. Math. Anal. 9(2): 201-247 (2015). DOI: 10.15352/bjma/09-2-15


The goal of this paper is to present characterizations for absolute continuity of representable positive functionals on general $^*$-algebras. From the results we give a new and very different proof to our recently published Lebesgue decomposition theorem for representable positive functionals. On unital $C^*$-algebras and measure algebras of compact groups further characterizations are included in the paper. As an application of our results, we answer Gudder's problem on the uniqueness of the Lebesgue decomposition in the case of commutative $^*$-algebras and measure algebras of compact groups. Another application to faithful positive functionals defined on the latter $^*$-algebras is also included.


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Zsolt Szűcs. "Absolute continuity of positive linear functionals." Banach J. Math. Anal. 9 (2) 201 - 247, 2015.


Published: 2015
First available in Project Euclid: 19 December 2014

zbMATH: 1347.46048
MathSciNet: MR3296115
Digital Object Identifier: 10.15352/bjma/09-2-15

Primary: 46L51
Secondary: 28A50 , 43A20 , 46C50 , 46L30 , 47A67

Keywords: Lebesgue type decompositions , measure algebras of compact groups , parallel sum , positive functionals , probability Radon measures

Rights: Copyright © 2015 Tusi Mathematical Research Group

Vol.9 • No. 2 • 2015
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