  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.

Operator Algebras