Comparison Properties of the Cuntz semigroup and applications to C*-algebras

TL;DR

This paper investigates comparison properties in the Cuntz semigroup category and their implications for C*-algebras, introducing a new property and exploring its relation to existing ones through examples.

Contribution

It introduces a new comparison property in the Cu category and relates it to the CFP and ω-comparison, providing examples that distinguish these properties.

Findings

Differences among comparison properties are demonstrated with examples.

The corona factorization property may allow both finite and infinite projections.

Rordam's simple, nuclear C*-algebra with finite and infinite projections lacks the CFP.

Abstract

We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{\o}rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.