Crossed products of Hilbert -bimodules

by countable discrete groups

Authors:
Tsuyoshi Kajiwara and Yasuo Watatani

Journal:
Proc. Amer. Math. Soc. **126** (1998), 841-851

MSC (1991):
Primary 46L05, 46L37, 46L55

DOI:
https://doi.org/10.1090/S0002-9939-98-04118-5

MathSciNet review:
1423344

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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.

**[Bl]**B. Blackadar,*K-Theory for operator algebras*, MSRI Publications**5**(1986). MR**88g:46082****[Ch]**M. Choda,*Duality for finite bipartite graphs*, Pacific J. Math.*(*with application to factors*)***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 -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*, 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 , 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*, 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 -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 -subalgebras*, Memoir Amer. Math. Soc.**83**, no. 424 (1990), 1-117. MR**90i:46104****[Y]**S. Yamagami,*Group symmetry in tensor categories*, preprint.

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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:
https://doi.org/10.1090/S0002-9939-98-04118-5

Received by editor(s):
May 15, 1996

Received by editor(s) in revised form:
September 10, 1996

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1998
American Mathematical Society