Nuclear dimension and $\mathcal {Z}$-stability of non-simple $\mathrm {C}^*$-algebras
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- by Leonel Robert and Aaron Tikuisis PDF
- Trans. Amer. Math. Soc. 369 (2017), 4631-4670 Request permission
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
- MR Author ID: 716339
- Email: lrobert@louisiana.edu
- Aaron Tikuisis
- Affiliation: Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom
- MR Author ID: 924851
- Email: a.tikuisis@abdn.ac.uk
- 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
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3632545