Representable linear functionals on partial *-algebras

TL;DR

This paper develops a GNS-like *-representation for partial *-algebras using representable linear functionals, introducing concepts like pre-core and singular forms, and analyzing their decompositions.

Contribution

It introduces a new GNS-like representation framework for partial *-algebras based on representable linear functionals and explores the structure of positive sesquilinear forms.

Findings

Decomposition of positive sesquilinear forms into ips and singular parts

Introduction of pre-core and singular form concepts

Construction of a GNS-like *-representation for partial *-algebras

Abstract

A GNS - like *-representation of a \pa\ $\A$ defined by certain representable linear functionals on $\A$ is constructed. The study of the interplay with the GNS construction associated with invariant positive sesquilinear forms (ips) leads to the notions of pre-core and of singular form. It is shown that a positive sesquilinear form with pre-core always decomposes into the sum of an ips form and a singular one.