The Order on Projections in C*-Algebras of Real Rank Zero

TL;DR

This paper investigates the order structure of projections in C*-algebras of real rank zero, establishing fundamental properties and characterizations related to lower bounds, greatest lower bounds, and the behavior under canonical homomorphisms.

Contribution

It provides new insights into the order properties of projections in real rank zero C*-algebras, including criteria for bounds and the preservation of least upper bounds.

Findings

The order on projections is separative.

Countable decreasing sequences of projections have equivalent lower bounds.

The preservation of least upper bounds by the canonical homomorphism occurs when the span of subspaces is closed.

Abstract

We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple product expressions of the projections in question. We also characterize the points at which the canonical homomorphism to the Calkin algebra preserves least upper bounds of countable collections of projections, namely that this occurs precisely when the span of the corresponding subspaces is closed.