The Cuntz Semigroup and Comparison of Open Projections

TL;DR

This paper explores the relationships between various comparison relations on positive elements in C*-algebras and their open projections, providing new insights into the structure of the Cuntz semigroup.

Contribution

It establishes equivalences between Cuntz comparison, Peligrad-Zsidó comparison, and Murray-von Neumann comparison via open projections, offering a new perspective on the Cuntz semigroup.

Findings

Cuntz comparison corresponds to a new comparison relation on open projections.

Murray-von Neumann comparison aligns with tracial comparison of positive elements.

Provides a unified framework linking different comparison notions in C*-algebras.

Abstract

We show that a number of naturally occurring comparison relations on positive elements in a C*-algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the C*-algebra. In particular we show that Cuntz comparison of positive elements corresponds to a comparison relation on open projections, that we call Cuntz comparison, and which is defined in terms of-and is weaker than-a comparison notion defined by Peligrad and Zsid\'o. The latter corresponds to a well-known comparison relation on positive elements defined by Blackadar. We show that Murray-von Neumann comparison of open projections corresponds to tracial comparison of the corresponding positive elements of the C*-algebra. We use these findings to give a new picture of the Cuntz semigroup.