Nearly relatively compact projections in operator algebras

TL;DR

This paper explores the concept of relative compactness of projections in operator algebras, unifying various inequalities through a numerical invariant and examining their relation to regularity and semicontinuity.

Contribution

It introduces a unified approach to understanding compactness and relative compactness of projections using a numerical invariant, linking inequalities, regularity, and semicontinuity.

Findings

A numerical invariant characterizes the distance from relative compactness.

Equivalence of inequalities to compactness and relative compactness.

Examples illustrating regularity and semicontinuity properties.

Abstract

Let A be a C*-algebra and A** its enveloping von Neumann algebra. C. Akemann suggested a kind of non-commutative topology in which certain projections in A** play the role of open sets. The adjectives "open", "closed", "compact", and "relatively compact" all can be applied to projections in A**. Two operator inequalities were used by Akemann in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in A**, but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, was used by the author. It turns out that the study of all three inequalities can be unified by considering a numerical invariant which is equivalent to the distance of a projection from the set of relatively compact projections. Since the subject concerns the relation between a projection and its closure, Tomita's concept of regularity of projections seems relevant, and some results and examples on regularity are also given. A few related results on semicontinuity are also included.