Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T09:31:30.563Z Has data issue: false hasContentIssue false
  • English
  • Français

HILBERT C*-BIMODULES OF FINITE INDEX AND APPROXIMATION PROPERTIES OF C*-ALGEBRAS

Part of: Selfadjoint operator algebras Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  17 October 2017

MARZIEH FOROUGH  and
MASSOUD AMINI
MARZIEH FOROUGH
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran e-mails: mforough86@gmail.com, mforough@ipm.ir
MASSOUD AMINI
Affiliation:
Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran School of Mathematics, Institute for Research in Fundamental Sciences, Tehran 19395-5746, Iran e-mails: mamini@modares.ac.ir, mamini@ipm.ir
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 A and B be arbitrary C*-algebras, we prove that the existence of a Hilbert A–B-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by A and B. For this, we first study the stability of the WEP, QWEP, and LLP under Morita equivalence of C*-algebras. We present examples of Hilbert A–B-bimodules, which are not of finite index, while such properties are shared between A and B. To this end, we study twisted crossed products by amenable discrete groups.

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46B28: Spaces of operators; tensor products; approximation properties 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 321 - 331
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. an Huef, A., Raeburn, I. and Williams, D. P., Properties preserved under Morita equivalnce of C*-algebras, Proc. Amer. Math. Soc. 135 (2006), 1495–1503.Google Scholar
2. Bedos, E., On actions of amenable groups on II 1-factors, J. Funct. Anal. 91 (1990), 404–414.Google Scholar
3. Bhattacharya, A. and Farenick, D., Crossed products of C*-algebras with the weak expectation property, New York J. Math. 19 (2013), 423–425.Google Scholar
4. Blecher, D. P., A new approach to Hilbert C*-modules, Math. Ann. 307 (1997), 253–290.Google Scholar
5. Brown, N. P., On quasidiagonal C*-algebras, Operator algebras and applications, 19–64, Adv. Stud. Pure Math., vol. 38 (Mathematical Society of Japan, Tokyo, 2004).Google Scholar
6. Brown, N. P. and Ozawa, N., C*-algebras and finite-dimensional approximation properties, Graduate Studies in Mathematics, vol. 88 (American Mathematical Society, Providence, 2008).Google Scholar
7. Dykemma, K. J. and Smith, R. R., The completely bounded approximation property for extended Cuntz-Pimsner algebra, Houston J. Math. 31 (2005), 829–840Google Scholar
8. Izumi, M., Inclusions of simple C*-algebras, J. Reine Angew. Math. 574 (2002), 97–138.Google Scholar
9. Jones, V. F. R., Index for subfactors, Invent. Math. 72 (1983), 1–25.Google Scholar
10. Kajiwara, T. and Watatani, Y., Crossed products of Hilbert C*-bimodules by countable discrete groups, Proc. Amer. Math. Soc. 126 (1998), 841–851.Google Scholar
11. Kajiwara, T. and Watatani, Y., Jones index theory by Hilbert C*-bimodules and the K-theory, Trans. Amer. Math. Soc. 352 (2000) 3429–3472.Google Scholar
12. Kajiwara, T., Pinzari, C. and Watatani, Y., Jones index theory by Hilbert C*-bimodules and its equivalence with conjugation theory, J. Funct. Anal. 215 (2004) 1–49.Google Scholar
13. Katsura, T., On C*-algebras associated with C*-correspondences, J. Funct. Anal. 217 (2004), 366–401.Google Scholar
14. Kirchberg, E., On nonsemisplit extensions, tensor products and exactness of group C*-algebras, Invent. Math. 112 (1993), 449–489.Google Scholar
15. Khoshkam, M., Hilbert C*-modules and conditional expectations on crossed products, J. Austral. Math. Soc., Ser. A 61 (1996), 106–118.Google Scholar
16. Frank, M. and Kirchberg, E., On conditional expectations of finite index, J. Oper. Theory 1 (1998), 87–111.Google Scholar
17. Osaka, H., Stable rank for inclusion of C*-algebras, Internat. J. Math. 19 (2008), 1011–1020.Google Scholar
18. Osaka, H., Kodaka, K. and Teruya, T., The Rokhlin property for inclusion of C*-algebras with finite Watatani index, Contemp. Math. 503 (2009), 177–195.Google Scholar
19. Osaka, H. and Teruya, T., Strongly self-absorbing property for inclusion of C*-algebras with a finite Watatani index, Trans. Amer. Math. Soc. 366 (2014), 1685–1702.Google Scholar
20. Ozawa, N., About the QWEP cojecture, Internat. J. Math. 15 (2004), 501–530.CrossRefGoogle Scholar
21. Pasnicu, C. and Phillips, N. C., Permanence properties for crossed products and fixed point algebras of finite groups, Trans. Amer. Math. Soc. 366 (2014), 4625–4648.Google Scholar
22. Popa, S., On the relative Dixmier property for inclusions of C*-algebras, J. Funct. Anal. 171 (2000), 139–154.Google Scholar
23. Rieffel, M., Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51–96.Google Scholar
24. Skalski, A. and Zacharias, J., On approximation properties of Pimsner algebras and crossed products by Hilbert bimodules, Rocky Mountain J. Math. 40 (2012), 609–625.Google Scholar
25. Watatani, Y., Index for C*-subalgebras, Memoirs Amer. Math. Soc., vol. 83 (American Mathematical Society, Providence, 1990).Google Scholar
26. Williams, D. P., Crossed products of C*-algebras, Mathematical Surveys and Monographs, vol. 134 (American Mathematical Society, Providence, 2007).Google Scholar