Operator algebras with contractive approximate identities

TL;DR

This paper explores the properties of operator algebras with contractive approximate identities, revealing inherent positivity and extending results from $C^*$-algebras to a broader class of operator algebras.

Contribution

It applies a recent theorem to establish new foundational results and generalizations for operator algebras with contractive approximate identities, connecting to noncommutative peak set theory.

Findings

Presence of significant positivity in operator algebras with contractive approximate identities

Generalization of several $C^*$-algebra results to broader operator algebras

Strengthening the theoretical foundation of noncommutative peak set theory

Abstract

We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for $C^*$-algebras, and we give many other applications.