Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 10, 2020

Decomposition rank of approximately subhomogeneous C*-algebras

George A. Elliott, Zhuang Niu, Luis Santiago and Aaron Tikuisis
From the journal Forum Mathematicum

Abstract

It is shown that every Jiang–Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. This settles a key direction of the Toms–Winter conjecture for simple approximately subhomogeneous C*-algebras. A key step in the proof is that subhomogeneous C*-algebras are locally approximated by a certain class of more tractable subhomogeneous algebras, namely a non-commutative generalization of the class of cell complexes. The result is applied, in combination with other recent results, to show classifiability of crossed product C*-algebras associated to minimal homeomorphisms with mean dimension zero.


Communicated by Christopher D. Sogge


Funding source: University of Wyoming

Award Identifier / Grant number: start-up grant

Funding source: Simons Foundation

Award Identifier / Grant number: collaboration grant

Funding source: University of Aberdeen

Award Identifier / Grant number: start-up grant

Funding statement: George A. Elliott has been supported by NSERC. Zhuang Niu has been supported by NSERC, a start-up grant from the University of Wyoming, and a Simons Foundation collaboration grant. Luis Santiago has been supported by the University of Toronto and a start-up grant from the University of Aberdeen. Aaron Tikuisis has been supported by NSERC and a start-up grant from the University of Aberdeen. All authors were supported by the Fields Institute through the “Thematic program on abstract harmonic analysis, Banach and operator algebras.” Work on this article advanced perceptibly at the EPSRC- and LMS-funded conference “Classification, structure, amenability, and regularity” in Glasgow, and at the BIRS workshop “Dynamics and -algebras: amenability and soficity.”

Acknowledgements

Aaron Tikuisis had numerous long discussions with Wilhelm Winter about the main problem solved in this article. We would like to thank Wilhelm Winter for these, and for comments on early versions of the article. We would also like to acknowledge Rob Archbold, Etienne Blanchard, Huaxin Lin, Chris Phillips, and Stuart White for discussions and comments that helped to shape this paper. Finally, we would like to thank the referees for helpful comments and suggestions.

References

[1] J. Bosa, N. P. Brown, Y. Sato, A. Tikuisis, S. White and W. Winter, Covering dimension of C*-algebras and 2-coloured classification, Mem. Amer. Math. Soc. 257 (2019), no. 1233. Search in Google Scholar

[2] L. G. Brown and G. K. Pedersen, Limits and C-algebras of low rank or dimension, J. Operator Theory 61 (2009), no. 2, 381–417. Search in Google Scholar

[3] N. P. Brown and W. Winter, Quasitraces are traces: a short proof of the finite-nuclear-dimension case, C. R. Math. Acad. Sci. Soc. R. Can. 33 (2011), no. 2, 44–49. Search in Google Scholar

[4] J. R. Carrión, Classification of a class of crossed product C-algebras associated with residually finite groups, J. Funct. Anal. 260 (2011), no. 9, 2815–2825. 10.1016/j.jfa.2011.02.002Search in Google Scholar

[5] J. Castillejos, S. Evington, A. Tikuisis, S. White and W. Winter, Nuclear dimension of simple C*-algebras, preprint (2019), https://arxiv.org/abs/1901.05853. 10.1007/s00222-020-01013-1Search in Google Scholar

[6] J. Castillejos and S. Evington, Nuclear dimension of simple stably projectionless C-algebras, preprint (2019), https://arxiv.org/abs/1901.11441; Anal. PDE, to appear. Search in Google Scholar

[7] S. R. Eilers, T. A. Loring and G. K. Pedersen, Stability of anticommutation relations: An application of noncommutative CW complexes, J. Reine Angew. Math. 499 (1998), 101–143. 10.1515/crll.1998.055Search in Google Scholar

[8] G. A. Elliott, An invariant for simple C*-algebras, Canadian Mathematical Society. 1945–1995, Vol. 3, Canadian Mathematical Society, Ottawa (1996), 61–90. Search in Google Scholar

[9] G. A. Elliott and D. E. Evans, The structure of the irrational rotation C*-algebra, Ann. of Math. (2) 138 (1993), no. 3, 477–501. 10.2307/2946553Search in Google Scholar

[10] G. A. Elliott, G. Gong, H. Lin and Z. Niu, The classification of simple separable unital 𝒵-stable locally ASH algebras, J. Funct. Anal. 272 (2017), no. 12, 5307–5359. 10.1016/j.jfa.2017.03.001Search in Google Scholar

[11] G. A. Elliott, G. Gong, H. Lin and Z. Niu, On the classification of simple amenable C*-algebras with finite decomposition rank, II, preprint (2015), https://arxiv.org/abs/1507.03437. Search in Google Scholar

[12] G. A. Elliott and Z. Niu, On the classification of simple amenable C*-algebras with finite decomposition rank, Operator Algebras and Their Applications, Contemp. Math. 671, American Mathematical Society, Providence (2016), 117–125. 10.1090/conm/671/13506Search in Google Scholar

[13] G. A. Elliott and Z. Niu, The C*-algebra of a minimal homeomorphism of zero mean dimension, Duke Math. J. 166 (2017), no. 18, 3569–3594. 10.1215/00127094-2017-0033Search in Google Scholar

[14] D. Enders, Subalgebras of finite codimension in semiprojective C*-algebras, Proc. Amer. Math. Soc. 145 (2017), no. 11, 4795–4805. 10.1090/proc/13620Search in Google Scholar

[15] R. Engelking, Dimension Theory, North-Holland Math. Libr. 19, North-Holland Publishing, Amsterdam, 1978. Search in Google Scholar

[16] T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and C*-crossed products, J. Reine Angew. Math. 469 (1995), 51–111. Search in Google Scholar

[17] G. Gong, H. Lin and Z. Niu, Classification of finite simple amenable 𝒵-stable C-algebras, preprint (2015), https://arxiv.org/abs/1501.00135. Search in Google Scholar

[18] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Search in Google Scholar

[19] S.-T. Hu, Theory of Retracts, Wayne State University Press, Detroit, 1965. Search in Google Scholar

[20] B. Jacelon, A simple, monotracial, stably projectionless C-algebra, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 365–383. 10.1112/jlms/jds049Search in Google Scholar

[21] E. Kirchberg and M. Rørdam, Purely infinite C*-algebras: Ideal-preserving zero homotopies, Geom. Funct. Anal. 15 (2005), no. 2, 377–415. 10.1007/s00039-005-0510-2Search in Google Scholar

[22] E. Kirchberg and W. Winter, Covering dimension and quasidiagonality, Internat. J. Math. 15 (2004), no. 1, 63–85. 10.1142/S0129167X04002119Search in Google Scholar

[23] H. Lin, Asymptotic unitary equivalence and classification of simple amenable C*-algebras, Invent. Math. 183 (2011), no. 2, 385–450. 10.1007/s00222-010-0280-9Search in Google Scholar

[24] H. Lin, Crossed products and minimal dynamical systems, J. Topol. Anal. 10 (2018), no. 2, 447–469. 10.1142/S1793525318500140Search in Google Scholar

[25] H. Lin and N. C. Phillips, Crossed products by minimal homeomorphisms, J. Reine Angew. Math. 641 (2010), 95–122. 10.1515/crelle.2010.029Search in Google Scholar

[26] Q. Lin, Analytic structure of transformation group C-algebras associated with minimal dynamical systems, preprint. Search in Google Scholar

[27] Q. Lin and N. C. Phillips, The structure of C-algebras of minimal diffeomorphisms, in preparation. Search in Google Scholar

[28] T. A. Loring, Lifting Solutions to Perturbing Problems in C*-Algebras, Fields Inst. Monogr. 8, American Mathematical Society, Providence, 1997. 10.1090/fim/008Search in Google Scholar

[29] H. Matui and Y. Sato, Decomposition rank of UHF-absorbing C*-algebras, Duke Math. J. 163 (2014), no. 14, 2687–2708. 10.1215/00127094-2826908Search in Google Scholar

[30] P. W. Ng and W. Winter, A note on subhomogeneous C-algebras, C. R. Math. Acad. Sci. Soc. R. Can. 28 (2006), no. 3, 91–96. Search in Google Scholar

[31] P. W. Ng and W. Winter, Nuclear dimension and the corona factorization property, Int. Math. Res. Not. IMRN 2010 (2010), no. 2, 261–278. 10.1093/imrn/rnp125Search in Google Scholar

[32] G. K. Pedersen, C-Algebras and Their Automorphism Groups, London Math. Soc. Monogr. 14, Academic Press, London, 1979. Search in Google Scholar

[33] N. C. Phillips, Cancellation and stable rank for direct limits of recursive subhomogeneous algebras, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4625–4652. 10.1090/S0002-9947-07-03849-4Search in Google Scholar

[34] N. C. Phillips, Recursive subhomogeneous algebras, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4595–4623. 10.1090/S0002-9947-07-03850-0Search in Google Scholar

[35] I. F. Putnam, The C*-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math. 136 (1989), no. 2, 329–353. 10.2140/pjm.1989.136.329Search in Google Scholar

[36] I. F. Putnam, On the K-theory of C*-algebras of principal groupoids, Rocky Mountain J. Math. 28 (1998), no. 4, 1483–1518. 10.1216/rmjm/1181071727Search in Google Scholar

[37] M. Rørdam, On the structure of simple C*-algebras tensored with a UHF-algebra. II, J. Funct. Anal. 107 (1992), no. 2, 255–269. 10.1016/0022-1236(92)90106-SSearch in Google Scholar

[38] M. Rørdam and W. Winter, The Jiang–Su algebra revisited, J. Reine Angew. Math. 642 (2010), 129–155. 10.1515/crelle.2010.039Search in Google Scholar

[39] L. Robert, Classification of inductive limits of 1-dimensional NCCW complexes, Adv. Math. 231 (2012), no. 5, 2802–2836. 10.1016/j.aim.2012.07.010Search in Google Scholar

[40] J. Rosenberg and C. Schochet, The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55 (1987), no. 2, 431–474. 10.1215/S0012-7094-87-05524-4Search in Google Scholar

[41] L. Santiago, Reduction of the dimension of nuclear C-algebras, preprint (2012), https://arxiv.org/abs/1211.7159. Search in Google Scholar

[42] Y. Sato, S. White and W. Winter, Nuclear dimension and 𝒵-stability, Invent. Math. 202 (2015), no. 2, 893–921. 10.1007/s00222-015-0580-1Search in Google Scholar

[43] K. R. Strung and W. Winter, Minimal dynamics and 𝒵-stable classification, Internat. J. Math. 22 (2011), no. 1, 1–23. 10.1142/S0129167X10006665Search in Google Scholar

[44] A. Tikuisis, High-dimensional 𝒵-stable AH algebras, J. Funct. Anal. 269 (2015), no. 7, 2171–2186. 10.1016/j.jfa.2015.05.013Search in Google Scholar

[45] A. Tikuisis, S. White and W. Winter, Quasidiagonality of nuclear C-algebras, Ann. of Math. (2) 185 (2017), no. 1, 229–284. Search in Google Scholar

[46] A. Tikuisis and W. Winter, Decomposition rank of 𝒵-stable C*-algebras, Anal. PDE 7 (2014), no. 3, 673–700. 10.2140/apde.2014.7.673Search in Google Scholar

[47] A. S. Toms, K-theoretic rigidity and slow dimension growth, Invent. Math. 183 (2011), no. 2, 225–244. 10.1007/s00222-010-0273-8Search in Google Scholar

[48] A. S. Toms and W. Winter, Minimal dynamics and the classification of C*-algebras, Proc. Natl. Acad. Sci. USA 106 (2009), no. 40, 16942–16943. 10.1073/pnas.0903629106Search in Google Scholar PubMed PubMed Central

[49] A. S. Toms and W. Winter, The Elliott conjecture for Villadsen algebras of the first type, J. Funct. Anal. 256 (2009), no. 5, 1311–1340. 10.1016/j.jfa.2008.12.015Search in Google Scholar

[50] A. S. Toms and W. Winter, Minimal dynamics and K-theoretic rigidity: Elliott’s conjecture, Geom. Funct. Anal. 23 (2013), no. 1, 467–481. 10.1007/s00039-012-0208-1Search in Google Scholar

[51] W. Winter, Decomposition rank of subhomogeneous C*-algebras, Proc. Lond. Math. Soc. (3) 89 (2004), no. 2, 427–456. 10.1112/S0024611504014716Search in Google Scholar

[52] W. Winter, On the classification of simple 𝒵-stable C*-algebras with real rank zero and finite decomposition rank, J. Lond. Math. Soc. (2) 74 (2006), no. 1, 167–183. 10.1112/S0024610706022903Search in Google Scholar

[53] W. Winter, Decomposition rank and 𝒵-stability, Invent. Math. 179 (2010), no. 2, 229–301. 10.1007/s00222-009-0216-4Search in Google Scholar

[54] W. Winter, Classifying crossed product C*-algebras, Amer. J. Math. 138 (2016), no. 3, 793–820. 10.1353/ajm.2016.0029Search in Google Scholar

Received: 2020-01-22
Revised: 2020-01-28
Published Online: 2020-03-10
Published in Print: 2020-07-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Scroll Up Arrow