Open projections in operator algebras II: Compact projections

TL;DR

This paper extends the theory of compact projections from C*-algebras to more general algebras, showing their relation to peak projections and establishing new noncommutative Urysohn lemmas.

Contribution

It introduces a generalized framework for compact projections, demonstrating their approximation by peak projections and characterizing them via state space faces.

Findings

Compact projections are decreasing limits of peak projections.

In separable cases, compact projections coincide with peak projections.

A projection is compact iff its associated face in the state space is weak* closed.

Abstract

We generalize some aspects of the theory of compact projections relative to a C*-algebra, to the setting of more general algebras. Our main result is that compact projections are the decreasing limits of `peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the associated face in the state space of the algebra is weak* closed.