Crossed products of Hilbert C*-bimodules by countable discrete groups

Kajiwara, Tsuyoshi; Watatani, Yasuo

doi:10.1090/S0002-9939-98-04118-5

Crossed products of Hilbert C$^\ast$-bimodules by countable discrete groups
HTML articles powered by AMS MathViewer

by Tsuyoshi Kajiwara and Yasuo Watatani PDF

Proc. Amer. Math. Soc. 126 (1998), 841-851 Request permission

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

Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867, DOI 10.1007/978-1-4613-9572-0
Marie Choda, Duality for finite bipartite graphs (with an application to $\textrm {II}_1$ factors), Pacific J. Math. 158 (1993), no. 1, 49–65. MR 1200828, DOI 10.2140/pjm.1993.158.49
Marie Choda and Hideki Kosaki, Strongly outer actions for an inclusion of factors, J. Funct. Anal. 122 (1994), no. 2, 315–332. MR 1276161, DOI 10.1006/jfan.1994.1071
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.
Raúl E. Curto, Paul S. Muhly, and Dana P. Williams, Cross products of strongly Morita equivalent $C^{\ast }$-algebras, Proc. Amer. Math. Soc. 90 (1984), no. 4, 528–530. MR 733400, DOI 10.1090/S0002-9939-1984-0733400-1
F. Combes, Crossed products and Morita equivalence, Proc. London Math. Soc. (3) 49 (1984), no. 2, 289–306. MR 748991, DOI 10.1112/plms/s3-49.2.289
Satoshi Goto, Orbifold construction for non-AFD subfactors, Internat. J. Math. 5 (1994), no. 5, 725–746. MR 1297414, DOI 10.1142/S0129167X9400036X
Masaki Izumi, Application of fusion rules to classification of subfactors, Publ. Res. Inst. Math. Sci. 27 (1991), no. 6, 953–994. MR 1145672, DOI 10.2977/prims/1195169007
Masaki Izumi, On type $\textrm {II}$ and type $\textrm {III}$ principal graphs of subfactors, Math. Scand. 73 (1993), no. 2, 307–319. MR 1269266, DOI 10.7146/math.scand.a-12473
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127
Tsuyoshi Kajiwara, Remarks on strongly Morita equivalent $C^*$-crossed products, Math. Japon. 32 (1987), no. 2, 257–260. MR 895546
G.G.Kasparov, K-theory, group $C^*$-algebras and higher signatures (Conspectus), Chernogolovka (1981).
Y. Katayama Takesaki’s duality on bipartite graph, preprint.
Yasuyuki Kawahigashi, On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors, J. Funct. Anal. 127 (1995), no. 1, 63–107. MR 1308617, DOI 10.1006/jfan.1995.1003
T. Kajiwara and S. Yamagami, Irreducible bimodules associated with crossed product algebras. II, Pacific J. Math. 171 (1995), no. 1, 209–229. MR 1362984, DOI 10.2140/pjm.1995.171.209
M. Khoshkam, Hilbert $C^*$-modules and conditional expectations on crossed products, J.Austral. Math. Soc. 61 (1996), 106–118.
T. Kajiwara and Y. Watatani, Jones index theory by Hilbert C${}^{*}$-bimodules and K-theory, preprint.
Hideki Kosaki, Automorphisms in the irreducible decomposition of sectors, Quantum and non-commutative analysis (Kyoto, 1992) Math. Phys. Stud., vol. 16, Kluwer Acad. Publ., Dordrecht, 1993, pp. 305–316. MR 1276299
Hideki Kosaki, Sector theory and automorphisms for factor-subfactor pairs, J. Math. Soc. Japan 48 (1996), no. 3, 427–454. MR 1389988, DOI 10.2969/jmsj/04830427
Hideki Kosaki and Shigeru Yamagami, Irreducible bimodules associated with crossed product algebras, Internat. J. Math. 3 (1992), no. 5, 661–676. MR 1189679, DOI 10.1142/S0129167X9200031X
Alexander Kumjian, Iain Raeburn, and Dana P. Williams, The equivariant Brauer groups of commuting free and proper actions are isomorphic, Proc. Amer. Math. Soc. 124 (1996), no. 3, 809–817. MR 1301034, DOI 10.1090/S0002-9939-96-03146-2
Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. MR 996454
Sorin Popa, On the classification of actions of amenable groups on subfactors, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 3, 295–299 (English, with English and French summaries). MR 1179723, DOI 10.1142/S0129167X10006343
Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255. MR 1278111, DOI 10.1007/BF02392646
Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811, DOI 10.24033/asens.1504
Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1
Marc A. Rieffel, Morita equivalence for $C^{\ast }$-algebras and $W^{\ast }$-algebras, J. Pure Appl. Algebra 5 (1974), 51–96. MR 367670, DOI 10.1016/0022-4049(74)90003-6
Marc A. Rieffel, Morita equivalence for operator algebras, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 285–298. MR 679708
Yasuo Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117. MR 996807, DOI 10.1090/memo/0424
S. Yamagami, Group symmetry in tensor categories, preprint.