The Cuntz semigroup and comparison of open projections

Eduard Ortega; Mikael Rørdam; Hannes Thiel

doi:10.1016/j.jfa.2011.02.017

The Cuntz semigroup and comparison of open projections

Eduard Ortega, Mikael Rørdam, Hannes Thiel

Department of Mathematical Sciences

32 Citations (Scopus)

Abstract

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.

Original language	English
Journal	Journal of Functional Analysis
Volume	260
Issue number	12
Pages (from-to)	3474-3493
Number of pages	20
ISSN	0022-1236
DOIs	https://doi.org/10.1016/j.jfa.2011.02.017
Publication status	Published - 15 Jun 2011

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AU - Ortega, Eduard

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AU - Thiel, Hannes

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N2 - 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.

AB - 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.

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DO - 10.1016/j.jfa.2011.02.017

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

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ER -

The Cuntz semigroup and comparison of open projections

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