Real positivity and approximate identities in Banach algebras

TL;DR

This paper extends the concept of positivity from operator algebras to broader classes of Banach algebras, aiming to generalize key $C^*$-algebraic results.

Contribution

It generalizes the notion of positivity and approximate identities from operator algebras to larger Banach algebra classes.

Findings

Extended positivity concepts to broader Banach algebras.

Generalized $C^*$-algebraic results to new algebra classes.

Provided new insights into approximate identities in Banach algebras.

Abstract

Blecher and Read have recently introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain $C^*$-algebraic results and theories to more general algebras. In the present paper we generalize some part of this, and some other facts, to larger classes of Banach algebras.