Commutativity of automorphisms of subfactors modulo inner automorphisms
HTML articles powered by AMS MathViewer
- by Satoshi Goto PDF
- Proc. Amer. Math. Soc. 124 (1996), 3391-3398 Request permission
Abstract:
We introduce a new algebraic invariant $\chi _{a}(M,N)$ of a subfactor $N \subset M$. We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant $\chi (M,N)$ introduced by Y. Kawahigashi. We also show that this group is contained in the center of $\operatorname {Out}(M,N)$ in many interesting examples such as quantum $SU(n)_{k}$ subfactors with level $k$ $(k \geq n+1)$, but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.References
- 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
- Alain Connes, Sur le théorème de Radon-Nikodym pour les poids normaux fidèles semi-finis, Bull. Sci. Math. (2) 97 (1973), 253–258 (1974) (French). MR 358375
- Alain Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 383–419. MR 394228, DOI 10.24033/asens.1295
- D. E. Evans & Y. Kawahigashi, Orbifold subfactors from Hecke algebras, Comm. Math. Phys. 165 (1994), 445–484.
- David E. Evans and Yasuyuki Kawahigashi, Subfactors and conformal field theory, Quantum and non-commutative analysis (Kyoto, 1992) Math. Phys. Stud., vol. 16, Kluwer Acad. Publ., Dordrecht, 1993, pp. 341–369. MR 1276304
- Satoshi Goto, Orbifold construction for non-AFD subfactors, Internat. J. Math. 5 (1994), no. 5, 725–746. MR 1297414, DOI 10.1142/S0129167X9400036X
- S. Goto, Symmetric flat connections triviality of Loi’s invariant and orbifold subfactors, to appear in Publ. RIMS Kyoto Univ.
- 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
- Yasuyuki Kawahigashi, Automorphisms commuting with a conditional expectation onto a subfactor with finite index, J. Operator Theory 28 (1992), no. 1, 127–145. MR 1259921
- 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
- Yasuyuki Kawahigashi, Centrally trivial automorphisms and an analogue of Connes’s $\chi (M)$ for subfactors, Duke Math. J. 71 (1993), no. 1, 93–118. MR 1230287, DOI 10.1215/S0012-7094-93-07105-0
- 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
- P. H. Loi, On automorphisms of subfactors, preprint, 1990.
- 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
- S. Popa, Classification of actions of discrete amenable groups on amenable subfactors of type II, preprint, 1992.
- Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255. MR 1278111, DOI 10.1007/BF02392646
- Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI 10.1007/BF01404457
- F. Xu, Orbifold construction in subfactors, Comm. Math. Phys. 166 (1994), 237–254.
- F. Xu, The flat part of non-flat orbifolds, 1993, to appear in Pac. J. of Math.
- Shigeru Yamagami, A note on Ocneanu’s approach to Jones’ index theory, Internat. J. Math. 4 (1993), no. 5, 859–871. MR 1245354, DOI 10.1142/S0129167X9300039X
- S. Yamagami, Modular theory for bimodules, J. Funct. Anal. 125 (1994), 327–357.
Additional Information
- Satoshi Goto
- Affiliation: Department of Mathematics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102, Japan
- Email: s-goto@hoffman.cc.sophia.ac.jp
- Received by editor(s): March 9, 1995
- Received by editor(s) in revised form: May 8, 1995
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3391-3398
- MSC (1991): Primary 46L37
- DOI: https://doi.org/10.1090/S0002-9939-96-03443-0
- MathSciNet review: 1340387