On the order structure of representable functionals

TL;DR

This paper explores the order structure of representable functionals on *-algebras, characterizing extreme points, infimum existence, and providing a Lebesgue decomposition approach analogous to positive operators.

Contribution

It introduces a new approach to analyze the order structure of representable functionals, including criteria for infimum existence and a Lebesgue decomposition framework.

Findings

Characterization of extreme points of order intervals.

A sufficient condition for the existence of infimum of two functionals.

A Lebesgue decomposition theory analogous to positive operators.

Abstract

The main purpose of this paper is to investigate some natural problems regarding the order structure of representable functionals on $^*$-algebras. We describe the extreme points of order intervals, and give a nontrivial sufficient condition to decide whether or not the infimum of two representable functionals exists. To this aim we offer a suitable approach to the Lebesgue decomposition theory, which is in complete analogy with the one developed by Ando in the context of positive operators. This tight analogy allows to invoke Ando's results to characterize uniqueness of the decomposition, and solve the infimum problem over certain operator algebras.