The Cuntz semigroup and comparison of open projections

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 Zsidó. 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.