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Nuclear dimension and $ \mathcal{Z}$-stability of non-simple $ \mathrm{C}^*$-algebras


Authors: Leonel Robert and Aaron Tikuisis
Journal: Trans. Amer. Math. Soc. 369 (2017), 4631-4670
MSC (2010): Primary 46L35; Secondary 46L80, 46L05, 46L06, 47L40, 46L85, 46L55
DOI: https://doi.org/10.1090/tran/6842
Published electronically: December 27, 2016
MathSciNet review: 3632545
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Abstract: We investigate the interplay of the following regularity properties for non-simple $ \mathrm C^*$-algebras: finite nuclear dimension, $ \mathcal Z$-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic regularity in the Cuntz semigroup, provided that known type I obstructions are avoided. We demonstrate how finite nuclear dimension can be used to study the structure of the central sequence algebra, by factorizing the identity map on the central sequence algebra, in a manner resembling the factorization arising in the definition of nuclear dimension.

Results about the central sequence algebra are used to attack the conjecture that finite nuclear dimension implies $ \mathcal {Z}$-stability, for sufficiently non-type I, separable $ \mathrm C^*$-algebras. We prove this conjecture in the following cases: (i) the $ \mathrm C^*$-algebra has no simple purely infinite ideals of quotients and its primitive ideal space has a basis of compact open sets, (ii) the $ \mathrm C^*$-algebra has no purely infinite quotients and its primitive ideal space is Hausdorff. In particular, this covers $ \mathrm C^*$-algebras with finite decomposition rank and real rank zero. Our results hold more generally for $ \mathrm C^*$-algebras with locally finite nuclear dimension which are $ (M,N)$-pure (a regularity condition of the Cuntz semigroup).


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Additional Information

Leonel Robert
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
Email: lrobert@louisiana.edu

Aaron Tikuisis
Affiliation: Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom
Email: a.tikuisis@abdn.ac.uk

DOI: https://doi.org/10.1090/tran/6842
Received by editor(s): June 3, 2014
Received by editor(s) in revised form: July 8, 2015
Published electronically: December 27, 2016
Additional Notes: The second-named author was partially supported by DFG (SFB 878) and an NSERC PDF
Article copyright: © Copyright 2016 American Mathematical Society