Covering dimension for nuclear $C^*$-algebras II
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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
- MR Author ID: 671014
- Email: wwinter@math.uni-muenster.de, wilhelm.winter@nottingham.ac.uk
- 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)
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 4143-4167
- MSC (2000): Primary 46L85, 46L35
- DOI: https://doi.org/10.1090/S0002-9947-09-04602-9
- MathSciNet review: 2500882