Positive combinations of projections in von Neumann algebras and purely infinite simple C*-algebras

TL;DR

This paper investigates which positive elements in operator algebras, specifically von Neumann and purely infinite simple C*-algebras, can be expressed as linear combinations or sums of projections with positive coefficients.

Contribution

It provides an overview and analysis of conditions under which positive elements can be represented as combinations of projections in these classes of algebras.

Findings

Characterization of positive elements as sums of projections

Conditions for positive elements to be linear combinations of projections

Insights into the structure of purely infinite simple C*-algebras

Abstract

We give an overview of the question: which positive elements in an operator algebra can be written as a linear combination of projections with positive coefficients. A special case of independent interest is the question of which positive elements can be written as a sum of finitely many projections. We focus on von Neumann algebras, on purely infinite simple C*-algebras, and on their associated multiplier algebras.