Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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).


References [Enhancements On Off] (What's this?)

  • [1] Ramon Antoine, Joan Bosa, Francesc Perera, and Henning Petzka, Geometric structure of dimension functions of certain continuous fields, J. Funct. Anal. 266 (2014), no. 4, 2403-2423. MR 3150165, https://doi.org/10.1016/j.jfa.2013.09.013
  • [2] Ramon Antoine, Joan Bosa, and Francesc Perera, Completions of monoids with applications to the Cuntz semigroup, Internat. J. Math. 22 (2011), no. 6, 837-861. MR 2812090, https://doi.org/10.1142/S0129167X11007057
  • [3] Bruce Blackadar and David Handelman, Dimension functions and traces on $ C^{\ast } $-algebras, J. Funct. Anal. 45 (1982), no. 3, 297-340. MR 650185, https://doi.org/10.1016/0022-1236(82)90009-X
  • [4] Bruce Blackadar, Leonel Robert, Aaron P. Tikuisis, Andrew S. Toms, and Wilhelm Winter, An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3657-3674. MR 2901228, https://doi.org/10.1090/S0002-9947-2012-05538-3
  • [5] Etienne Blanchard and Eberhard Kirchberg, Non-simple purely infinite $ C^*$-algebras: the Hausdorff case, J. Funct. Anal. 207 (2004), no. 2, 461-513. MR 2032998, https://doi.org/10.1016/j.jfa.2003.06.008
  • [6] Marius Dadarlat, Fiberwise $ KK$-equivalence of continuous fields of $ \mathrm C^*$-algebras, J. K-Theory 3 (2009), no. 2, 205-219. MR 2496447 (2010j:46122)
  • [7] Marius Dadarlat and Andrew S. Toms, $ \mathcal Z$-stability and infinite tensor powers of $ \mathrm C^*$-algebras, Adv. Math. 220 (2009), no. 2, 341-366. MR 2466419 (2010c:46132)
  • [8] George A. Elliott and Andrew S. Toms, Regularity properties in the classification program for separable amenable $ C^*$-algebras, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 229-245. MR 2383304, https://doi.org/10.1090/S0273-0979-08-01199-3
  • [9] Ilijas Farah, Leonel Robert, and Aaron Tikuisis, Model theory of central sequence algebras, work in progress.
  • [10] K. R. Goodearl and D. Handelman, Rank functions and $ K_{0}$ of regular rings, J. Pure Appl. Algebra 7 (1976), no. 2, 195-216. MR 0389965 (52 #10794)
  • [11] Uffe Haagerup, Quasitraces on exact $ C^*$-algebras are traces, C. R. Math. Acad. Sci. Soc. R. Can. 36 (2014), no. 2-3, 67-92. Circulated in manuscript form in 1991 (English, with English and French summaries). MR 3241179
  • [12] Ilan Hirshberg, Mikael Rørdam, and Wilhelm Winter, $ \mathcal {C}_0(X)$-algebras, stability and strongly self-absorbing $ C^*$-algebras, Math. Ann. 339 (2007), no. 3, 695-732. MR 2336064, https://doi.org/10.1007/s00208-007-0129-8
  • [13] Bhishan Jacelon, $ \mathcal {Z}$-stability, finite dimensional tracial boundaries and continuous rank functions, Münster J. Math. 6 (2013), no. 2, 583-594. MR 3148224
  • [14] Eberhard Kirchberg, On nonsemisplit extensions, tensor products and exactness of group $ \mathrm C^*$-algebras, Invent. Math. 112 (1993), no. 3, 449-489. MR 1218321 (94d:46058)
  • [15] Eberhard Kirchberg, Central sequences in $ C^*$-algebras and strongly purely infinite algebras, Operator Algebras: The Abel Symposium 2004, Abel Symp., vol. 1, Springer, Berlin, 2006, pp. 175-231. MR 2265050, https://doi.org/10.1007/978-3-540-34197-0_10
  • [16] Eberhard Kirchberg and Mikael Rørdam, Infinite non-simple $ C^*$-algebras: absorbing the Cuntz algebras $ \mathcal {O}_\infty $, Adv. Math. 167 (2002), no. 2, 195-264. MR 1906257, https://doi.org/10.1006/aima.2001.2041
  • [17] Eberhard Kirchberg and Mikael Rørdam, Central sequence $ C^*$-algebras and tensorial absorption of the Jiang-Su algebra, J. Reine Angew. Math. 695 (2014), 175-214. MR 3276157, https://doi.org/10.1515/crelle-2012-0118
  • [18] Hiroki Matui and Yasuhiko Sato, Strict comparison and $ \mathcal {Z}$-absorption of nuclear $ C^*$-algebras, Acta Math. 209 (2012), no. 1, 179-196. MR 2979512, https://doi.org/10.1007/s11511-012-0084-4
  • [19] Hiroki Matui and Yasuhiko Sato, Decomposition rank of UHF-absorbing $ \mathrm C^*$-algebras, Duke Math. J. 163 (2014), no. 14, 2687-2708.
  • [20] D. McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. 21 (1970), 443-461.
  • [21] Norio Nawata, Picard groups of certain stably projectionless $ \rm C^*$-algebras, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 161-180. MR 3092263, https://doi.org/10.1112/jlms/jdt013
  • [22] Eduard Ortega, Francesc Perera, and Mikael Rørdam, The corona factorization property, stability, and the Cuntz semigroup of a $ C^\ast $-algebra, Int. Math. Res. Not. IMRN 1 (2012), 34-66. MR 2874927
  • [23] Gert K. Pedersen, $ \mathrm C^*$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1979. MR 548006 (81e:46037)
  • [24] Leonel Robert, Nuclear dimension and $ n$-comparison, Münster J. Math. 4 (2011), 65-71. MR 2869254
  • [25] Leonel Robert and Mikael Rørdam, Divisibility properties for $ C^*$-algebras, Proc. Lond. Math. Soc. (3) 106 (2013), no. 6, 1330-1370. MR 3072284, https://doi.org/10.1112/plms/pds082
  • [26] Mikael Rørdam, On the structure of simple $ C^*$-algebras tensored with a UHF-algebra. II, J. Funct. Anal. 107 (1992), no. 2, 255-269. MR 1172023, https://doi.org/10.1016/0022-1236(92)90106-S
  • [27] Mikael Rørdam, A simple $ C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109-142. MR 2020420, https://doi.org/10.1007/BF02392697
  • [28] Mikael Rørdam, The stable and the real rank of $ \mathcal Z$-absorbing $ \mathrm C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065-1084. MR 2106263 (2005k:46164)
  • [29] M. Rørdam and E. Størmer, Classification of nuclear $ C^*$-algebras. Entropy in operator algebras, Encyclopaedia of Mathematical Sciences, vol. 126, Operator Algebras and Non-commutative Geometry, 7, Springer-Verlag, Berlin, 2002. MR 1878881
  • [30] Mikael Rørdam and Wilhelm Winter, The Jiang-Su algebra revisited, J. Reine Angew. Math. 642 (2010), 129-155. MR 2658184
  • [31] Yasuhiko Sato, Trace spaces of simple nuclear $ \mathrm C^*$-algebras with finite-dimensional extreme boundary, 2012. arXiv preprint 1209.3000.
  • [32] Yasuhiko Sato, Stuart White, and Wilhelm Winter, Nuclear dimension and $ \mathcal {Z}$-stability, Invent. Math. 202 (2015), no. 2, 893-921. MR 3418247, https://doi.org/10.1007/s00222-015-0580-1
  • [33] Adam Sierakowski, The ideal structure of reduced crossed products, Münster J. Math. 3 (2010), 237-261. MR 2775364
  • [34] Gábor Szabó, The Rokhlin dimension of topological $ \mathbb{Z}^m$-actions, 2013. Preprint.
  • [35] Aaron Tikuisis, Nuclear dimension, $ \mathcal {Z}$-stability, and algebraic simplicity for stably projectionless $ C^\ast $-algebras, Math. Ann. 358 (2014), no. 3-4, 729-778. MR 3175139, https://doi.org/10.1007/s00208-013-0951-0
  • [36] Aaron Tikuisis and Wilhelm Winter, Decomposition rank of $ {\mathcal {Z}}$-stable $ \rm C^*$-algebras, Anal. PDE 7 (2014), no. 3, 673-700. MR 3227429, https://doi.org/10.2140/apde.2014.7.673
  • [37] Andrew S. Toms, On the classification problem for nuclear $ \mathrm C^*$-algebras, Ann. of Math. (2) 167 (2008), no. 3, 1029-1044. MR 2415391 (2009g:46119)
  • [38] Andrew S. Toms, Stuart White, and Wilhelm Winter, $ \mathcal {Z}$-stability and finite-dimensional tracial boundaries, Int. Math. Res. Not. IMRN 10 (2015), 2702-2727. MR 3352253, https://doi.org/10.1093/imrn/rnu001
  • [39] Andrew S. Toms and Wilhelm Winter, $ \mathcal {Z}$-stable ASH algebras, Canad. J. Math. 60 (2008), no. 3, 703-720. MR 2414961, https://doi.org/10.4153/CJM-2008-031-6
  • [40] Jesper Villadsen, Simple $ \mathrm C^*$-algebras with perforation, J. Funct. Anal. 154 (1998), no. 1, 110-116. MR 1616504 (99j:46069)
  • [41] Dan Voiculescu, A note on quasi-diagonal $ \mathrm C^*$-algebras and homotopy, Duke Math. J. 62 (1991), no. 2, 267-271. MR 1104525, https://doi.org/10.1215/S0012-7094-91-06211-3
  • [42] Wilhelm Winter, Decomposition rank of subhomogeneous $ \mathrm C^*$-algebras, Proc. London Math. Soc. (3) 89 (2004), no. 2, 427-456. MR 2078703 (2005d:46121)
  • [43] Wilhelm Winter, Decomposition rank and $ \mathcal {Z}$-stability, Invent. Math. 179 (2010), no. 2, 229-301. MR 2570118, https://doi.org/10.1007/s00222-009-0216-4
  • [44] Wilhelm Winter, Strongly self-absorbing $ \mathrm C^*$-algebras are $ \mathcal Z$-stable, J. Noncommut. Geom. 5 (2011), no. 2, 253-264. MR 2784504 (2012e:46132)
  • [45] Wilhelm Winter, Nuclear dimension and $ \mathcal {Z}$-stability of pure $ \mathrm C^*$-algebras, Invent. Math. 187 (2012), no. 2, 259-342.
  • [46] Wilhelm Winter and Joachim Zacharias, Completely positive maps of order zero, Münster J. Math. 2 (2009), 311-324. MR 2545617
  • [47] Wilhelm Winter and Joachim Zacharias, The nuclear dimension of $ \mathrm C^*$-algebras, Adv. Math. 224 (2010), no. 2, 461-498. MR 2609012 (2011e:46095)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L35, 46L80, 46L05, 46L06, 47L40, 46L85, 46L55

Retrieve articles in all journals with MSC (2010): 46L35, 46L80, 46L05, 46L06, 47L40, 46L85, 46L55


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

American Mathematical Society