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Regularity properties in the classification program for separable amenable C$ ^*$-algebras


Authors: George A. Elliott and Andrew S. Toms
Journal: Bull. Amer. Math. Soc. 45 (2008), 229-245
MSC (2000): Primary 46L35; Secondary 46L80
DOI: https://doi.org/10.1090/S0273-0979-08-01199-3
Published electronically: February 12, 2008
MathSciNet review: 2383304
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Abstract: We report on recent progress in the program to classify separable amenable C$ ^*$-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and $ \mathcal{Z}$-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.


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Additional Information

George A. Elliott
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 2E4
Email: elliott@math.toronto.edu

Andrew S. Toms
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada, M3J 1P3
Email: atoms@mathstat.yorku.ca

DOI: https://doi.org/10.1090/S0273-0979-08-01199-3
Keywords: {C}$^*$-algebras, classification
Received by editor(s): April 20, 2007
Received by editor(s) in revised form: October 1, 2007
Published electronically: February 12, 2008
Additional Notes: This work was partially supported by the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2008 American Mathematical Society

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