On Characters and Pure States of *-Algebras

TL;DR

This paper explores the relationship between characters and pure states in *-algebras, introducing the concept of abstract O*-algebras to generalize and extend existing results to both commutative and non-commutative cases.

Contribution

It introduces the notion of abstract O*-algebras and provides sufficient conditions for characters to be pure states in a broad algebraic setting.

Findings

Characters of commutative unital *-algebras are pure states.

Sufficient conditions are established for the converse in certain *-algebras.

Many concepts extend to non-commutative *-algebras.

Abstract

It is easy to see that every character (i.e. unital *-homomorphism to the complex numbers) of a commutative unital associative *-algebra is a pure state (i.e. extreme point in the convex set of all normalized positive linear functionals). This article gives sufficient conditions for the converse to be true as well. In order to formulate these results together with similar ones, e.g. for locally convex *-algebras, the notion of an abstract O*-algebra (unital associative *-algebra with an order defined by positive linear functionals) is introduced. Many concepts and intermediary results discussed here also apply to the non-commutative case.