Order theory and interpolation in operator algebras

TL;DR

This paper advances the theory of positivity and order in operator algebras with contractive approximate identities, extending C*-algebra concepts to broader contexts and exploring various applications in noncommutative topology and interpolation.

Contribution

It develops a new notion of positivity and ordering in operator algebras, providing foundational results and applications that extend C*-algebra theories to more general operator algebras.

Findings

Established foundational facts about positivity in operator algebras.

Applied the theory to noncommutative topology and peak interpolation.

Results are applicable even without approximate identities.

Abstract

We continue our study of operator algebras with and contractive approximate identities (cais). In earlier papers we have 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. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator algebra and the C*-algebra it generates. In much of this it is not necessary that the algebra have an approximate identity. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.