The Cuntz Semigroup and Comparison of Open Projections

Details The Cuntz Semigroup and Comparison of Open Projections The Cuntz Semigroup and Comparison of Open Projections

2010-08-20

arxiv.org/abs/1008.3497v1

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.

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