A characterisation of C*-algebras through positivity of functionals

TL;DR

This paper characterizes C*-algebras among unital involutive Banach algebras by a positivity condition on functionals that attain their norm at the identity element.

Contribution

It provides a new characterization of C*-algebras based on the positivity of certain norm-attaining functionals, expanding understanding of their structural properties.

Findings

A unital involutive Banach algebra is a C*-algebra if all norm-attaining functionals at the identity are positive.

The characterization links positivity of functionals to the algebra's C*-structure.

The result offers a functional-analytic criterion for identifying C*-algebras.

Abstract

We show that a unital involutive Banach algebra, with identity of norm one and continuous involution, is a C*-algebra, with the given involution and norm, if every continuous linear functional attaining its norm at the identity is positive.