Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-28T07:45:36.358Z Has data issue: false hasContentIssue false
  • English
  • Français

All Irrational Extended Rotation Algebras are AF Algebras

Published online by Cambridge University Press:  20 November 2018

George A. Elliott  and
Zhuang Niu
George A. Elliott
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 e-mail: elliott@math.toronto.edu
Zhuang Niu
Affiliation:
Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA e-mail: zniu@uwyo.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\theta \,\in \,\left[ 0,\,1 \right]$ be any irrational number. It is shown that the extended rotation algebra ${{B}_{\theta }}$ introduced by the authors in J. Reine Angew. Math. 665(2012), pp. 1–71, is always an $\text{AF}$ algebra.

Keywords

46L0546L3546L5546L80irrational rotation algebraextended irrational rotation algebraAF-embedding.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 4 , 01 August 2015 , pp. 810 - 826
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Blackadar, B., K-theory for Operator Algebras. Second edition, Math. Sci. Res. Inst. Publ. 5, Cambridge University Press, Cambridge, 1998.Google Scholar
[2] Blackadar, B., Kumjian, A., and Rørdam, M., Approximately central matrix units and the structure of noncommutative tori. K-Theory 6(1992), 267–284.http://dx.doi.org/10.1007/BF00961466 Google Scholar
[3] Brown, L. G., Extensions of AF-algebras: the projection lifting problem. In: Operator algebras and applications Symposia in Pure Math. 38, Providence, RI, 1981, 175–716.Google Scholar
[4] Effros, E. G., Handelman, D. E., and Shen, C. L., Dimension groups and their affine representations. Amer. J. Math. 102(1980), 385–407.http://dx.doi.org/10.2307/2374244 Google Scholar
[5] Elliott, G. A., Automorphisms determined by multipliers on ideals of a C*-algebra. J. Funct. Anal. 23(1976), 1–10.http://dx.doi.org/10.1016/0022-1236(76)90054-9 Google Scholar
[6] Elliott, G. A. and Evans, D. E., The structure of the irrational rotation C*-algebra. Ann. of Math. (2) 138(1993), 477–501. http://dx.doi.org/10.2307/2946553 Google Scholar
[7] Elliott, G. A. and Niu, Z., A canonical Pimsner-Voiculescu embedding. In: Oberwolfach Reports, Mathematisches Forschungsinstitut Oberwolfach, 43, European Mathematical Society Publishing House, Zürich, 2013, pp. 2438–2440. http://dx.doi.org/10.4171/OWR/2013/43 Google Scholar
[8] Elliott, G. A. and Niu, Z., Extended rotation algebras: adjoining spectral projections to rotation algebras. J. Reine Angew. Math. 665(2012), 1–71. http://dx.doi.org/10.1515/CRELLE.2011.112 Google Scholar
[9] Elliott, G. A. and Niu, Z., Remarks on the Pimsner-Voiculescu embedding. In: Operator Algebra and Dynamics (Nordforsk Network Closing Conference, Faroe Islands. May 2012), Springer Proceedings in Mathematics & Statistics 58, 2013, pp. 131–140.Google Scholar
[10] Gong, G., Jiang, X., and Su, H., Obstructions to 풵-stability for unital simple C*-algebras. Canad. Math. Bull. 43(2000), 418–426.http://dx.doi.org/10.4153/CMB-2000-050-1 Google Scholar
[11] Lin, H., Localizing the Elliott conjecture at strongly self-absorbing C*-algebras, II. J. Reine Angew. Math. 2014(2014) no. 692, 233–243.http://dx.doi.org/10.1515/crelle-2012-0182 Google Scholar
[12] Lin, H. and Niu, Z., Lifting KK-elements, asymptotic unitary equivalence and classification of simple C*-algebras. Adv. Math. 219(2008), 1729–1769.http://dx.doi.org/10.1016/j.aim.2008.07.011 Google Scholar
[13] Matui, H. and Sato, Y., Strict comparison and 풵-absorption of nuclear C*-algebras. Acta Math. 209(2012), 179–196. http://dx.doi.org/10.1007/s11511-012-0084-4 Google Scholar
[14] Matui, H. and Sato, Y., Decomposition rank of UHF-absorbing C*-algebras. Duke Math. J., to appear. arxiv:1303.4371.Google Scholar
[15] Phillips, N. C., Large subalgebras. Preprint, 2014.Google Scholar
[16] Pimsner, M. and Voiculescu, D., Imbedding the irrational rotation C*-algebra into an AF-algebra. J. Operator Theory 4(1980), 201–210.Google Scholar
[17] Putnam, I. F., The C*-algebras associated with minimal homeomorphisms of the Cantor set. Pacific J. Math. 136(1989):329–353. http://dx.doi.org/10.2140/pjm.1989.136.329 Google Scholar
[18] Sato, Y., Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary. Preprint, 2012. arxiv:1209.3000.Google Scholar
[19] Thomsen, K., On the KK-theory and the E-theory of amalgamated free products of C*-algebras. J. Funct. Anal. 201(2003), 30–56. http://dx.doi.org/10.1016/S0022-1236(03)00084-3 Google Scholar
[20] Winter, W., Localizing the Elliott conjecture at strongly self-absorbing C*-algebras. J. Reine Angew. Math. 2014(2014) no. 692, 193–231.http://dx.doi.org/10.1515/crelle-2012-0082 Google Scholar