Skip to main content Accessibility help
×
×
Home

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

  • MARZIEH FOROUGH (a1) and MASSOUD AMINI (a2) (a3)

Abstract

Let A and B be arbitrary C*-algebras, we prove that the existence of a Hilbert AB-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 AB-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.

Copyright

References

Hide All
1. an Huef, A., Raeburn, I. and Williams, D. P., Properties preserved under Morita equivalnce of C*-algebras, Proc. Amer. Math. Soc. 135 (2006), 14951503.
2. Bedos, E., On actions of amenable groups on II 1-factors, J. Funct. Anal. 91 (1990), 404414.
3. Bhattacharya, A. and Farenick, D., Crossed products of C*-algebras with the weak expectation property, New York J. Math. 19 (2013), 423425.
4. Blecher, D. P., A new approach to Hilbert C*-modules, Math. Ann. 307 (1997), 253290.
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).
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).
7. Dykemma, K. J. and Smith, R. R., The completely bounded approximation property for extended Cuntz-Pimsner algebra, Houston J. Math. 31 (2005), 829840
8. Izumi, M., Inclusions of simple C*-algebras, J. Reine Angew. Math. 574 (2002), 97138.
9. Jones, V. F. R., Index for subfactors, Invent. Math. 72 (1983), 125.
10. Kajiwara, T. and Watatani, Y., Crossed products of Hilbert C*-bimodules by countable discrete groups, Proc. Amer. Math. Soc. 126 (1998), 841851.
11. Kajiwara, T. and Watatani, Y., Jones index theory by Hilbert C*-bimodules and the K-theory, Trans. Amer. Math. Soc. 352 (2000) 34293472.
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) 149.
13. Katsura, T., On C*-algebras associated with C*-correspondences, J. Funct. Anal. 217 (2004), 366401.
14. Kirchberg, E., On nonsemisplit extensions, tensor products and exactness of group C*-algebras, Invent. Math. 112 (1993), 449489.
15. Khoshkam, M., Hilbert C*-modules and conditional expectations on crossed products, J. Austral. Math. Soc., Ser. A 61 (1996), 106118.
16. Frank, M. and Kirchberg, E., On conditional expectations of finite index, J. Oper. Theory 1 (1998), 87111.
17. Osaka, H., Stable rank for inclusion of C*-algebras, Internat. J. Math. 19 (2008), 10111020.
18. Osaka, H., Kodaka, K. and Teruya, T., The Rokhlin property for inclusion of C*-algebras with finite Watatani index, Contemp. Math. 503 (2009), 177195.
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), 16851702.
20. Ozawa, N., About the QWEP cojecture, Internat. J. Math. 15 (2004), 501530.
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), 46254648.
22. Popa, S., On the relative Dixmier property for inclusions of C*-algebras, J. Funct. Anal. 171 (2000), 139154.
23. Rieffel, M., Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 5196.
24. Skalski, A. and Zacharias, J., On approximation properties of Pimsner algebras and crossed products by Hilbert bimodules, Rocky Mountain J. Math. 40 (2012), 609625.
25. Watatani, Y., Index for C*-subalgebras, Memoirs Amer. Math. Soc., vol. 83 (American Mathematical Society, Providence, 1990).
26. Williams, D. P., Crossed products of C*-algebras, Mathematical Surveys and Monographs, vol. 134 (American Mathematical Society, Providence, 2007).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed