Operator algebras with contractive approximate identities II

TL;DR

This paper advances the understanding of operator algebras, focusing on their structural properties, noncommutative peak interpolation, and the extension of compact projection theory, even for algebras lacking approximate identities.

Contribution

It provides new results on the existence of contractive approximate identities and extends the theory of compact projections in operator algebras.

Findings

An operator algebra has a contractive approximate identity iff the span of elements with positive real part is dense.

Extended the theory of compact projections to the most general case.

Made progress in noncommutative peak interpolation theory.

Abstract

We make several contributions to our recent program investigating structural properties of algebras of operators on a Hilbert space. For example, we make substantial contributions to the noncommutative peak interpolation program begun by Hay and the first author, Hay and Neal. Another sample result: an operator algebra has a contractive approximate identity iff the linear span of the elements with positive real part is dense. We also extend the theory of compact projections to the most general case. Despite the title, our algebras are often allowed to have no approximate identity.