Lebesgue decomposition for representable functionals on $^*$-algebras

TL;DR

This paper extends Lebesgue decomposition concepts to representable functionals on $^*$-algebras, generalizing previous results and establishing mutual absolute continuity of the absolutely continuous parts.

Contribution

It introduces a Lebesgue-type decomposition for representable functionals on $^*$-algebras, broadening the scope of earlier work on Banach $^*$-algebras.

Findings

Decomposition into absolutely continuous and singular parts

Mutual absolute continuity of the absolutely continuous components

Generalization of Gudder's results to non-unital $^*$-algebras

Abstract

We present a Lebesgue-type decomposition for a representable functional on a $^*$-algebra into absolutely continuous and singular parts with respect to an other. This generalizes the corresponding results of S. P. Gudder for unital Banach $^*$-algebras. We also prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.