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

Institut for Matematiske Fag

32 Citationer (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.

Originalsprog	Engelsk
Tidsskrift	Journal of Functional Analysis
Vol/bind	260
Udgave nummer	12
Sider (fra-til)	3474-3493
Antal sider	20
ISSN	0022-1236
DOI	https://doi.org/10.1016/j.jfa.2011.02.017
Status	Udgivet - 15 jun. 2011

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10.1016/j.jfa.2011.02.017

Citationsformater

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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 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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T1 - The Cuntz semigroup and comparison of open projections

AU - Ortega, Eduard

AU - Rørdam, Mikael

AU - Thiel, Hannes

PY - 2011/6/15

Y1 - 2011/6/15

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.

U2 - 10.1016/j.jfa.2011.02.017

DO - 10.1016/j.jfa.2011.02.017

M3 - Journal article

SN - 0022-1236

VL - 260

SP - 3474

EP - 3493

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 12

ER -

The Cuntz semigroup and comparison of open projections

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