Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Jones index theory by Hilbert C${}^{*}$-bimodules and K-theory

Author(s): Tsuyoshi Kajiwara; Yasuo Watatani
Journal: Trans. Amer. Math. Soc. 352 (2000), 3429-3472.
MSC (2000): Primary 46L08, 46L80
Posted: April 14, 2000
MathSciNet review: 1624182
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: In this paper we introduce the notion of Hilbert ${\mathrm{C}}^{*}$-bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert ${\mathrm{C}}^{*}$-bimodules and show that tensor products of minimal bimodules are also minimal. For an $A$-$A$ bimodule which is compatible with a trace on a unital ${\mathrm{C}}^{*}$-algebra $A$, its dimension (square root of Jones index) depends only on its $KK$-class. Finally, we show that the dimension map transforms the Kasparov products in $KK(A,A)$ to the product of positive real numbers, and determine the subring of $KK(A,A)$ generated by the Hilbert ${\mathrm{C}}^{*}$-bimodules for a ${\mathrm{C}}^{*}$-algebra generated by Jones projections.

References:

[1]
B.Blackadar, $K$-theory for operator algebras, MSRI Publ., vol. 5, Springer-Verlag, 1986. MR 88g:46082

[2]
L.G.Brown, P.Green and M.A.Rieffel, Stable isomorphism and strong Morita equivalence of C${}^*$-algebras, Pacific J. Math. 71 (1977), 349-363. MR 57:3866

[3]
L.G.Brown,J.A.Mingo and N.T.Shen, Quasi-multipliers and embeddings of Hilbert C${}^*$-bimodules, Canad. J. Math. 46 (1994), 1150-1174. MR 95k:46091

[4]
M.Choda and H.Kosaki, Strongly outer actions for an inclusion of factors, J. Funct. Anal. 122 (1994), 315-332. MR 96b:46085

[5]
A.Connes, Noncommutative Geometry, Academic Press (1994). MR 95j:46063

[6]
V.Deaconu, C${}^*$-Algebras associated with Branched Coverings, preprint.

[7]
V.Deaconu, A path model for circle algebras, J. Operator Theory 34 (1995), 57-89. MR 96m:46113

[8]
Y.Denizeau and J.-F.Havet, Correspondences d'indice fini I, J. Operator Theory 32 (1994), 111-156. MR 96e:46081

[9]
Y.Denizeau and J.-F.Havet, Correspondences d'indice fini II, Topics in Operator Theory, Operator Algebras and Applications, Rom. Acad. Bucharest (1995), 49-79. MR 97i:46110

[10]
S.Doplicher,R.Haag and J.E.Roberts, Local observables and particle statistics I, Commun.Math.Phys. 23 (1971), 199-230. MR 45:6316

[11]
S.Doplicher,R.Haag and J.E.Roberts, Local observables and particle statistics II, Commun.Math.Phys. 35 (1974), 49-85. MR 48:13060

[12]
S.Doplicher and J.E.Roberts, A new duality theory for compact groups, Invent.Math. 98 (1989), 157-218. MR 90k:22005

[13]
M.Enomoto, M.Fujii, H.Takehana and Y.Watatani, Automorphisms on Cuntz Algebras, II, Math.Japonica 24 (1979), 463-468. MR 81i:46086b

[14]
D.Evans and Y.Kawahigashi, On Ocneanu's theory of asymptotic inclusions for subfactors, topological quantum field theories and quantum double, Internat. J. Math. 6 (1995), 205-228. MR 96d:46080

[15]
R.Exel, A Fredholm operator approach to Morita equivalence, K-Theory 7 (1993), 285-308. MR 94h:46107

[16]
F.M.Goodman, P.de la Harpe and V.F.R. Jones, Coxeter Graphs and Towers of Algebras, MSRI Publications, vol. 14, Springer-Verlag, 1989. MR 91c:46082

[17]
P.Ghez, R.Lima and J.E.Roberts, W${}^{*}$-categories, Pac.J.Math. 120(1985) 79-109. MR 87g:46091

[18]
J.-F.Havet, Esperence conditionelle minimele, J.Operator Theory 24 (1991), 33-55. MR 92a:46073

[19]
F.Hiai, Minimizing indices of conditional expectation onto a subfactor, Publ.RIMS 24 (1988), 673-678. MR 90a:46157

[20]
M.Izumi, Application of fusion rules to classification of subfactors, Publ.RIMS.Kyoto Univ. 27 (1991), 953-994. MR 93b:46121

[21]
M.Izumi, Goldman's type theorem for index 3, Publ.RIMS.Kyoto.Univ. 28 (1992), 833-843. MR 94m:46093

[22]
M.Izumi, Subalgebras of infinite C${}^*$-algebras with finite Watatani Indices I. Cuntz algebras, Comm.Math.Phys. 155 (1995), 157-182. MR 94e:46104

[23]
V.F.R.Jones, Index for subfactors, Inv. Math. 72 (1983), 1-25. MR 84d:46097

[24]
L.Kadison, On split, separable subalgebras with counitality condition, Hokkaido Math. J. 24 (1995), 527-549. MR 98b:46081

[25]
C.Kassel, Quantum Groups, Graduate Texts in Math., vol. 155, Springer-Verlag, 1995. MR 96e:17041

[26]
G.G.Kasparov, Hilbert C${}^*$-module: Theorem of Steinspring and Voiculescu, J.Operator Theory 4 (1980), 133-150. MR 82b:46074

[27]
G.G.Kasparov, The operator K-functor and extensions of C${}^*$-algebras, Math.USSR-Izv 16 (1981), 513-572. MR 81m:58075

[28]
S.Kawakami and Y Watatani, The multiplicativity of the minimal index of simple C${}^*$-algebras, Proc. Amer. Math. Soc. 123 (1995), 2809-2813. MR 95k:46093

[29]
M.Khoshkam, Hilbert C${}^*$-modules and conditional expectations on crossed products, J. Austral. Math. Soc. (Series A) 61 (1996), 106-118. MR 97i:46100

[30]
K.Kodaka, Endomorphisms of certain irrational rotation C${}^*$-algebras, Illinois J. Math. 36 (1992), 643-658. MR 94b:46086

[31]
H.Kosaki, Automorphisms in the irreducible decomposition of sectors, Quantum and Non-commutative Analysis, Kluwer Academic (1993), 305-316. MR 95e:46073

[32]
H.Kosaki and R.Longo, A remark on the minimal index of subfactors, J.Funct.Anal. 107 (1992), 458-470. MR 93i:46108

[33]
H.Kosaki and S.Yamagami, Irreducible bimodules associated with crossed product algebras, Internat. J. Math. 3 (1992), 661-676. MR 94f:46087

[34]
R.Longo, Index of subfactors and statistics of quantum fields I, Commun.Math.Phys. 126 (1989), 245-155. MR 91c:46097

[35]
R.Longo, Index of subfactors and statistics of quantum fields II, Commun.Math.Phys. 130 (1990), 285-309. MR 91j:46083

[36]
R.Longo, Minimal index and braided subfactors, J.Funct.Anal. 107 (1992), 458-470. MR 93i:46109

[37]
A.Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications,London Math.Soc. Lect.Notes.Series 2 (1988), 119-172. MR 91k:46068

[38]
A.Ocneanu, Quantized symmetry, differential geometry of finite graphs and classification of subfactors (Note recorded by Y.Kawahigashi), University of Tokyo Seminor Notes 45 (1991).

[39]
W.Paschke, Inner product modules over ${\mathrm{B}}^{*}$-algebras, Trans. AMS. 182 (1973), 443-468. MR 50:8087

[40]
C.Peligrad and W.Szymanski, Saturated Actions of Finite Dimensional Hopf *-algebras on C${}^*$-Algebras, Math. Scand. 75 (1994), 217-239. MR 96b:46097

[41]
M.Pimsner and S.Popa, Entropy and index for subfactors, Ann.Sc.Ec.Norm.Sup. 19 (1985), 57-106. MR 87m:46120

[42]
S.Popa, Correspondences, preprint.

[43]
M.A.Rieffel, Induced representations of C${}^*$-algebras, Adv.Math. 13 (1974), 176-257. MR 50:5489

[44]
M.A.Rieffel, Morita equivalence for C${}^*$-algebras and W${}^{*}$ -algebras, J.Pure Appl. Algebra 5 (1974), 51-96. MR 51:3912

[45]
M.A.Rieffel, Morita equivalence for operator algebras, Proc Symp. in Pure Math. 38, 285-298. MR 84k:46045

[46]
V.S.Sunder, $II_{1}$-factor, their bimodules and hypergroups, Trans.Amer.Math.Soc. 330 (1992), 227-256. MR 92f:46076

[47]
Y.Watatani, Index for C${}^*$-subalgebras, Memoir AMS 424 (1990). MR 90i:46104

[48]
S.Yamagami, A note on Ocneanu's approach to Jones index theory, Intnat.J.Math. 4 (1993), 859-871. MR 95f:46114

[49]
S.Yamagami, On unitary representation theory of compact quantum groups, Comm. Math.Phys. 167 (1995), 509-529. MR 96a:46114

[50]
S.Yamagami, Frobenius reciprocity in tensor categories, preprint (1993).

[51]
S.Yamagami, Modular theory for bimodules, J.Funct.Anal 125 (1994), 327-357. MR 96h:46100

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L08, 46L80

Retrieve articles in all Journals with MSC (2000): 46L08, 46L80

Additional Information:

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

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

DOI: 10.1090/S0002-9947-00-02392-8
PII: S 0002-9947(00)02392-8
Keywords: Hilbert C${}^{*}$-bimodule, K-theory, Jones index, subfactor
Received by editor(s): April 28, 1995
Received by editor(s) in revised form: March 18, 1998
Posted: April 14, 2000
Additional Notes: Partially supported by the Grants--in--aid for Scientific Research, The Ministery of Education
Dedicated: Dedicated to the 60th birthday of Professor Takeshi Hirai
Copyright of article: Copyright 2000, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia