Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Crossed products of Hilbert $\mathrm{C}^{\ast}$-bimodules by countable discrete groups

Author(s): Tsuyoshi Kajiwara; Yasuo Watatani
Journal: Proc. Amer. Math. Soc. 126 (1998), 841-851.
MSC (1991): Primary 46L05, 46L37, 46L55
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We introduce a notion of crossed products of Hilbert C${}^{*}$-bimodules by countable discrete groups and mainly study the case of finite groups following Jones index theory. We give a sufficient condition such that the crossed product bimodule is irreducible. We have a bimodule version of Takesaki-Takai duality. We show the categorical structures when the action is properly outer, and give some example of this construction concerning the orbifold constructions.

References:

[Bl]
B. Blackadar, K-Theory for operator algebras, MSRI Publications 5 (1986). MR 88g:46082

[Ch]
M. Choda, Duality for finite bipartite graphs (with application to ${\mathrm{II}}_{1}$ factors), Pacific J. Math. 158 (1993), 49-65. MR 94e:46113

[CK]
M. Choda and H. Kosaki, Strongly outer actions for an inclusion of factors, J. Funct. Anal. 122 (1994), 315-332. MR 96b:46085

[CKRW]
D. Crocker, A. Kumjian, I. Raeburn and D. P. Williams, An equivariant Brauer group and actions of group $C^*$-algebras, J. Funct. Anal. 146 (1997), 171-184.

[CMW]
R. E. Curto, P. S. Muhly and D. P. Williams, Cross products of strongly Morita equivalent C${}^*$-algebras, Proc. AMS. 90 (1984), 528-530. MR 85i:46083

[Co]
F. Combes, Crossed products and Morita equivalence, Proc. London Math. Soc. 49 (1984), 289-306. MR 86c:46081

[Go]
S.Goto, Orbifold construction for Non-AFD subfactors, Internat. J. Math. 5 (1994), 725-746. MR 95h:46094

[I1]
M. Izumi, Application of fusion rules to classification of subfactors, Publ. RIMS Kyoto Univ. 27 (1991), 953-994. MR 93b:46121

[I2]
M.Izumi, On type II and type III principal graphs of subfactors, Math. Scand. 73 (1993), 307-319. MR 95a:46084

[Jo]
V. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. MR 84d:46097

[Kaj]
T. Kajiwara, Remarks on strongly Morita equivalent C${}^*$-crossed products, Math. Japon. 32 (1987), 257-260. MR 89h:46095

[Kas]
G.G.Kasparov, K-theory, group C${}^*$-algebras and higher signatures (Conspectus), Chernogolovka (1981).

[Kat]
Y. Katayama Takesaki's duality on bipartite graph, preprint.

[Kaw]
Y. Kawahigashi, On flatness of Ocneanu's connections on the Dynkin diagrams and classification of subfactors, J. Funct. Annal. 127 (1995), 63-107. MR 95j:46075

[KaY]
T. Kajiwara and S. Yamagami, Irreducible bimodules associated with crossed product algebras $II$, Pac. J. Math. 171 (1995), 209-229. MR 96m:46124

[Kh]
M. Khoshkam, Hilbert C${}^*$-modules and conditional expectations on crossed products, J.Austral. Math. Soc. 61 (1996), 106-118. CMP 96:16

[KW]
T. Kajiwara and Y. Watatani, Jones index theory by Hilbert C${}^{*}$-bimodules and K-theory, preprint.

[Ko1]
H. Kosaki, Automorphisms in the irreducible decompositions of sectors, Quantum and Non Commutative Analysis, Kluwer Academic (1993), 305-316. MR 95e:46073

[Ko2]
H. Kosaki, Sector theory and automorphisms for factor - subfactor pairs, J. Math. Soc. Japan 48 (1996), 427-454. MR 97e:46083

[KoY]
H. Kosaki and S. Yamagami, Irreducible bimodules associated with crossed product algebras, Int. Math. J. (1992), 661-676. MR 94f:46087

[KRW]
A. Kumjian, I. Raeburn and D. P. Williams, The equivariant Brauer groups of commuting free and proper actions are isomorphic, Proc. Amer. Math. Soc. 124 (1996), 809-817. MR 96f:46107

[Oc]
A. Ocneanu, Quantized groups, string algebras, and Galois theory for algebra , in Operator Algebras and Applications. Vol $II$, London Mathematical Lecture Note Series Vol. 136, Cambridge Univ. Press, 1988, 119-172. MR 91k:46068

[Po1]
S. Popa, On the classification of actions of amenable groups on subfactors, C.R. Acad. Sc. Paris. 315 (1992), 295-299. MR 93i:46110

[Po2]
S. Popa, Classification of amenable subfactor of type $II$, Acta Math. 172 (1994), 163-255. MR 95f:46105

[PP]
M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Scient. Ec. Norm. Sup. 19 (1986), 57-106. MR 87m:46120

[Ri1]
M. A. Rieffel, Induced representations of C${}^*$-algebras, Adv. Math. 13 (1974), 176-257. MR 50:5489

[Ri2]
M. A. Rieffel, Morita equivalence for C${}^*$-algebras and ${\mathrm{W}}^{*}$-algebras, J. Pure Appl. Algebras 5 (1974), 51-96. MR 51:3912

[Ri3]
M. A. Rieffel, Morita equivalence for operator algebras, Proc. Sympos. Pure Math. Amer. Math. Soc. 38, Part1 (1982), 285-298. MR 84k:46045

[Wa]
Y. Watatani, Index for $C^{*}$-subalgebras, Memoir Amer. Math. Soc. 83, no. 424 (1990), 1-117. MR 90i:46104

[Y]
S. Yamagami, Group symmetry in tensor categories, preprint.

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46L05, 46L37, 46L55

Retrieve articles in all Journals with MSC (1991): 46L05, 46L37, 46L55

Additional Information:

Tsuyoshi Kajiwara
Affiliation: Department of Environmental and Mathematical Sciences, Okayama University, Tsushima, Okayama 700, Japan
Email: kajiwara@math.ems.okayama-4.ac.jp

Yasuo Watatani
Affiliation: Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka, 810 Japan
Email: watatani@rc.kyush-u.ac.jp

DOI: 10.1090/S0002-9939-98-04118-5
PII: S 0002-9939(98)04118-5
Received by editor(s): May 15, 1996
Received by editor(s) in revised form: September 10, 1996
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1998, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google