Mathematics – Operator Algebras
Scientific paper
2010-08-20
Mathematics
Operator Algebras
Scientific paper
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.
Ortega Eduard
Rordam Mikael
Thiel Hannes
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