Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-27T00:47:21.524Z Has data issue: false hasContentIssue false
  • English
  • Français

A CHARACTERIZATION OF SATURATED C*-ALGEBRAIC BUNDLES OVER FINITE GROUPS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  04 June 2010

KAZUNORI KODAKA  and
TAMOTSU TERUYA
KAZUNORI KODAKA*
Affiliation:
Department of Mathematical Sciences, Faculty of Science, Ryukyu University, Nishihara-cho, Okinawa 903-0213, Japan (email: kodaka@math.u-ryukyu.ac.jp)
TAMOTSU TERUYA
Affiliation:
Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
*
✉For correspondence; e-mail: kodaka@math.u-ryukyu.ac.jp
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 be a unital C*-algebra. Let (B,E) be a pair consisting of a unital C*-algebra B containing A as a C*-subalgebra with a unit that is also the unit of B, and a conditional expectation E from B onto A that is of index-finite type and of depth 2. Let B1 be the C*-basic construction induced by (B,E). In this paper, we shall show that any such pair (B,E) satisfying the conditions that A′∩B=ℂ1 and that A′∩B1 is commutative is constructed by a saturated C*-algebraic bundle over a finite group. Furthermore, we shall give a necessary and sufficient condition for B to be described as a twisted crossed product of A by its twisted action of a finite group under the condition that A′∩B1 is commutative.

Keywords

conditional expectationsC*-algebraic bundlesdepth 2initial C*-algebrasWatatani indextwisted actionstwisted crossed products

MSC classification

Secondary: 46L08: $C^*$-modules
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 3 , June 2010 , pp. 363 - 383
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Blackadar, B., K-theory for Operator Algebras, 2nd edn Mathematical Sciences Research Institute Publication, 5 (Cambridge University Press, Cambridge, 1998).Google Scholar
[2]Brown, L. G., ‘Stable isomorphism of hereditary subalgebras’, Pacific J. Math. 71 (1977), 335–348.CrossRefGoogle Scholar
[3]Brown, L. G., Green, P. and Rieffel, M. A., ‘Stable isomorphism and strong Morita equivalence of C *-algebra’, Pacific J. Math. 71 (1977), 349–368.CrossRefGoogle Scholar
[4]Busby, R. C., ‘Double centralizers and extensions of C *-algebras’, Trans. Amer. Math. Soc. 132 (1968), 79–99.Google Scholar
[5]Exel, R., ‘Amenability for Fell bundles’, J. Reine Angew. Math. 492 (1997), 41–73.Google Scholar
[6]Fell, J. M. G. and Doran, R. S., Representations of *-Algebras, Locally Compact Groups and Banach*-Algebraic Bundles, Pure and Applied Mathematics, 125, 126 (Academic Press, Boston, MA, 1988).Google Scholar
[7]Goodman, F., de la Harpe, P. and Jones, V. F. R., Coxeter Graphs and Towers of Algebras, Mathematical Sciences Research Institute Publication, 14 (Springer, New York, 1989).CrossRefGoogle Scholar
[8]Izumi, M., ‘Inclusions of simple C *-algebras’, J. Reine Angew. Math. 547 (2002), 97–138.Google Scholar
[9]Jensen, K. K. and Thomsen, K., Elements of KK-theory (Birkhäuser, Basel, 1991).CrossRefGoogle Scholar
[10]Jones, V. F. R., Actions of Finite Groups on the Hyperfinite Type II 1 Factor, Memoirs of the American Mathematical Society, 237 (American Mathematical Society, Providence, RI, 1980).Google Scholar
[11]Jones, V. F. R., ‘Index for subfactors’, Invent. Math. 72 (1983), 1–25.CrossRefGoogle Scholar
[12]Jeong, J. A., Osaka, H., Phillips, N. C. and Teruya, T., ‘Cancellation for inclusions of C *-algebras of finite depth’, Indiana Univ. Math. J. 58 (2009), 1537–1564.CrossRefGoogle Scholar
[13]Kajiwara, T. and Watatani, Y., ‘Jones index theory for Hilbert C *-bimodules and K-theory’, Trans. Amer. Math. Soc. 352 (2000), 3429–3472.CrossRefGoogle Scholar
[14]Kodaka, K., ‘Full projections, equivalence bimodules and automorphisms of stable algebras of unital C *-algebras’, J. Operator Theory 37 (1997), 357–369.Google Scholar
[15]Kodaka, K. and Teruya, T., ‘Involutive equivalence bimodules and inclusions of C *-algebras with Watatani index 2’, J. Operator Theory 57 (2007), 3–18.Google Scholar
[16]Kosaki, H., ‘Characterization of crossed product (properly infinite case)’, Pacific J. Math. 137 (1989), 159–167.CrossRefGoogle Scholar
[17]Landstad, M. B., Phillips, J., Raeburn, I. and Surtherland, C., ‘Representations of crossed products by coactions and principal bundles’, Trans. Amer. Math. Soc. 299 (1987), 747–784.CrossRefGoogle Scholar
[18]Ocneanu, A., ‘Quantized groups, string algebras, and Galois theory’, in: Operator Algebras and Applications, Vol. II, London Mathematical Society Lecture Note Series, 136 (Cambridge University Press, Cambridge, 1988).Google Scholar
[19]Ocneanu, A., ‘Quantum symmetry, differential geometry of finite graphs and classification of subfactors’, University of Tokyo Seminar Notes (recorded by Y. Kawahigashi), 1991.Google Scholar
[20]Osaka, H. and Teruya, T., ‘Topological stable rank of inclusions of unital C *-algebras’, Internat. J. Math. 17 (2006), 19–34.CrossRefGoogle Scholar
[21]Osaka, H. and Teruya, T., ‘Stable rank of depth 2 inclusions of C *-algebras’, C. R. Math. Acad. Sci. Soc. R. Can. 29 (2007), 28–32.Google Scholar
[22]Packer, J. and Raeburn, I., ‘Twisted crossed products of C *-algebras’, Math. Proc. Cambridge Philos. Soc. 106 (1989), 293–311.CrossRefGoogle Scholar
[23]Pedersen, G. K., C *-Algebras and their Automorphism Groups (Academic Press, Amsterdam, 1979).Google Scholar
[24]Phillips, J. and Raeburn, I., ‘Automorphisms of C *-algebras and second Čech cohomology’, Indiana Univ. Math. J. 29 (1980), 799–822.CrossRefGoogle Scholar
[25]Popa, S., ‘Classification of subfactors: reduction to commuting squares’, Invent. Math. 101 (1990), 19–43.CrossRefGoogle Scholar
[26]Popa, S., ‘Classification of amenable subfactors of type II’, Acta Math. 172 (1994), 352–445.CrossRefGoogle Scholar
[27]Quigg, J. C., ‘Duality for reduced twisted crossed products of C *-algebras’, Indiana Univ. Math. J. 35 (1986), 549–571.CrossRefGoogle Scholar
[28]Quigg, J. C., ‘Full and reduced coactions’, Math. Proc. Cambridge Philos. Soc. 116 (1994), 435–450.CrossRefGoogle Scholar
[29]Quigg, J. C., ‘Discrete C *-coactions and C *-algebraic bundles’, J. Aust. Math. Soc. (Series A) 60 (1996), 204–221.CrossRefGoogle Scholar
[30]Rieffel, M. A., ‘Actions of finite groups on C *-algebras’, Math. Scand. 47 (1980), 157–176.CrossRefGoogle Scholar
[31]Rieffel, M. A., ‘Dimension and stable rank in the K-theory of C *-algebras’, Proc. London Math. Soc. 46 (1983), 301–333.CrossRefGoogle Scholar
[32]Sutherland, C. E., ‘Cohomology and extensions of von Neumann algebras. I, II’, Publ. Res. Inst. Math. Sci. 16 (1980), 105–133 and 135–174.CrossRefGoogle Scholar
[33]Szymański, W., ‘Finite index subfactors and Hopf algebra crossed products’, Proc. Amer. Math. Soc. 120 (1994), 519–528.CrossRefGoogle Scholar
[34]Teruya, T., ‘A characterization of normal extensions for subfactors’, Proc. Amer. Math. Soc. 120 (1994), 781–783.CrossRefGoogle Scholar
[35]Watatani, Y., Index for C *-subalgebras, Memoirs of the American Mathematical Society, 424 (American Mathematical Society, Providence, RI, 1990).CrossRefGoogle Scholar