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# Comparison Properties of the Cuntz semigroup and applications to C*-algebras

Published:2017-05-25
Printed: Feb 2018
• Joan Bosa,
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, G12 8QW, Glasgow, UK
• Henning Petzka,
Fraunhofer Institute for Intelligent Analysis and Information Systems IAIS, Schloss Birlinghoven, 53757 Sankt Augustin, Germany
 Format: LaTeX MathJax PDF

## Abstract

We study comparison properties in the category $\mathrm{Cu}$ aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{\o}rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.
 Keywords: classification of C*-algebras, cuntz semigroup
 MSC Classifications: 46L35 - Classifications of $C^*$-algebras 06F05 - Ordered semigroups and monoids [See also 20Mxx] 46L05 - General theory of $C^*$-algebras 19K14 - $K_0$ as an ordered group, traces

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