The Projection Calculus

TL;DR

This paper introduces tools for manipulating projections in C*-algebras, providing new proofs and strengthening classical results, especially for real rank zero algebras, and exploring their order properties.

Contribution

It develops new tools for projections in C*-algebras and applies them to improve proofs and results related to real rank zero algebras.

Findings

Short proofs of projection homotopy results

Strengthened classical results on liftings and pure states

Analysis of order properties of projections

Abstract

We develop some tools for manipulating and constructing projections in C*-algebras. These are then applied to give short proofs of some standard projection homotopy results, as well as strengthen some fundamental classical results for C*-algebras of real rank zero, specifically on liftings, excising pure states and Kadison's transitivity theorem. Lastly, we investigate some order properties of the set of projections in C*-algebras of real rank zero, building on our previous work.