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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Covering dimension for nuclear $ C^*$-algebras II


Author: Wilhelm Winter
Journal: Trans. Amer. Math. Soc. 361 (2009), 4143-4167
MSC (2000): Primary 46L85, 46L35
Published electronically: March 17, 2009
MathSciNet review: 2500882
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Abstract: The completely positive rank is an analogue of the topological covering dimension, defined for nuclear $ C^*$-algebras via completely positive approximations. These may be thought of as simplicial approximations of the algebra, which leads to the concept of piecewise homogeneous maps and a notion of noncommutative simplicial complexes.

We introduce a technical variation of completely positive rank and show that the two theories coincide in many important cases. Furthermore, we analyze some of their properties; in particular we show that both theories behave nicely with respect to ideals and that they coincide with the covering dimension of the spectrum for certain continuous trace $ C^*$-algebras.


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Additional Information

Wilhelm Winter
Affiliation: Mathematisches Institut der Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
Address at time of publication: School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
Email: wwinter@math.uni-muenster.de, wilhelm.winter@nottingham.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04602-9
PII: S 0002-9947(09)04602-9
Keywords: $C^*$-algebras, covering dimension
Received by editor(s): November 21, 2001
Received by editor(s) in revised form: October 11, 2005, and July 19, 2007
Published electronically: March 17, 2009
Additional Notes: The author was supported by EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280) and Deutsche Forschungsgemeinschaft (SFB 478)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.