Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Subalgebras of finite codimension in semiprojective $ C^*$-algebras


Author: Dominic Enders
Journal: Proc. Amer. Math. Soc. 145 (2017), 4795-4805
MSC (2010): Primary 46L05
DOI: https://doi.org/10.1090/proc/13620
Published electronically: May 26, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that semiprojectivity of a $ C^*$-algebra is preserved when passing to $ C^*$-subalgebras of finite codimension. In particular, any pullback of two semiprojective $ C^*$-algebras over a finite-dimensional $ C^*$-algebra is again semiprojective.


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

  • [Bla85] Bruce Blackadar, Shape theory for $ C^\ast $-algebras, Math. Scand. 56 (1985), no. 2, 249-275. MR 813640
  • [Bla04] Bruce Blackadar, Semiprojectivity in simple $ C^*$-algebras, Operator algebras and applications, Adv. Stud. Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 1-17. MR 2059799
  • [Cun87] Joachim Cuntz, A new look at $ KK$-theory, $ K$-Theory 1 (1987), no. 1, 31-51. MR 899916
  • [Dad94] Marius Dădărlat, Shape theory and asymptotic morphisms for $ C^*$-algebras, Duke Math. J. 73 (1994), no. 3, 687-711. MR 1262931
  • [Dad09] Marius Dadarlat, Continuous fields of $ C^*$-algebras over finite dimensional spaces, Adv. Math. 222 (2009), no. 5, 1850-1881. MR 2555914
  • [EK15] S. Eilers and T. Katsura, Semiprojectivity and properly infinite projections in graph $ C^*$-algebras, preprint, 2015, arXiv:1512.07277v1.
  • [ELP98] Søren Eilers, Terry A. Loring, and Gert K. Pedersen, Stability of anticommutation relations: an application of noncommutative CW complexes, J. Reine Angew. Math. 499 (1998), 101-143. MR 1631120
  • [ELP99] Søren Eilers, Terry A. Loring, and Gert K. Pedersen, Morphisms of extensions of $ C^*$-algebras: pushing forward the Busby invariant, Adv. Math. 147 (1999), no. 1, 74-109. MR 1725815
  • [End13] D. Enders, On the structure of certain classes of semiprojective $ C^*$-algebras, PhD thesis, WWU Münster, 2013.
  • [End14] Dominic Enders, A characterization of semiprojectivity for subhomogeneous $ C^*$-algebras, Doc. Math. 21 (2016), 987-1049. MR 3548139
  • [ERR13] S. Eilers, G. Restorff, and E. Ruiz, Strong classification of extensions of classifiable $ C^*$-algebras, preprint, 2013, arXiv:1301.7695v1.
  • [GLN15] G. Gong, H. Lin and Z. Niu, Classification of finite simple amenable $ \mathcal {Z}$-stable $ C^*$-algebras, preprint, 2015, arXiv:1501.00135v6.
  • [Lor96] Terry A. Loring, Stable relations. II. Corona semiprojectivity and dimension-drop $ C^*$-algebras, Pacific J. Math. 172 (1996), no. 2, 461-475. MR 1386627
  • [Lor97a] Terry A. Loring, Lifting solutions to perturbing problems in $ C^*$-algebras, Fields Institute Monographs, vol. 8, American Mathematical Society, Providence, RI, 1997. MR 1420863
  • [Lor97b] T. A. Loring, Perturbation questions in the Cuntz picture of $ K$-theory, $ K$-Theory 11 (1997), no. 2, 161-193. MR 1444287
  • [LP98] Terry A. Loring and Gert K. Pedersen, Projectivity, transitivity and AF-telescopes, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4313-4339. MR 1616003
  • [Ped79] Gert K. Pedersen, $ C^{\ast } $-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
  • [Sør12] A. P. W. Sørensen. On a counterexample to a conjecture by Blackadar, Operator Algebra and Dynamics, Springer Proceedings in Mathematics & Statistics, Vol. 58, 2013.
  • [Zek89] Richard Zekri, A new description of Kasparov's theory of $ C^*$-algebra extensions, J. Funct. Anal. 84 (1989), no. 2, 441-471. MR 1001468

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46L05

Retrieve articles in all journals with MSC (2010): 46L05


Additional Information

Dominic Enders
Affiliation: Westfälische Wilhelms-Universität, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
Email: d.enders@uni-muenster.de

DOI: https://doi.org/10.1090/proc/13620
Received by editor(s): July 20, 2016
Received by editor(s) in revised form: December 5, 2016
Published electronically: May 26, 2017
Additional Notes: This work was supported by the SFB 878 \itshape Groups, Geometry and Actions and the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92)
Communicated by: Adrian Ioana
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society