Open Projections and Suprema in the Cuntz Semigroup

TL;DR

This paper offers a new, concise proof for the existence of suprema in the Cuntz semigroup by utilizing open projections, showing that the supremum of countable open projections remains open and relates to hereditary subalgebras.

Contribution

It introduces a simplified proof for suprema in the Cuntz semigroup using open projections, advancing the understanding of their structure in C*-algebras.

Findings

Suprema of countable open projections are open.

Suprema correspond to hereditary sub-C*-algebras.

The proof simplifies previous approaches.

Abstract

We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated by Ortega, Rordam and Thiel. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary sub-C*-algebra of A.