Commutativity of automorphisms

of subfactors modulo inner automorphisms

Author:
Satoshi Goto

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new algebraic invariant of a subfactor . 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 introduced by Y. Kawahigashi. We also show that this group is contained in the center of in many interesting examples such as quantum subfactors with level , but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.

**[CK]**M. Choda & H. Kosaki,*Strongly outer actions for an inclusion of factors*, J. Funct. Anal.**122**(1994), 315--332. MR**96b:46085****[C1]**A. Connes,*Sur le theoreme de Radon-Nikodym pour les poids normauxfideles semi-finis*, Bull. Sci. Math.**97**(1973), 253--258. MR**50:10841****[C2]**A. Connes,*Outer conjugacy classes of automorphisms of factors*, Ann. Sci. Ec. Norm. Sup. 4me serie, t.8 (1975), 383--420. MR**52:15031****[EK1]**D. E. Evans & Y. Kawahigashi,*Orbifold subfactors from Hecke algebras*, Comm. Math. Phys.**165**(1994), 445--484. CMP**95:03****[EK2]**D. E. Evans & Y. Kawahigashi,*Subfactors and conformal field theory*, Quantum and non-commutative analysis, Kluwer Academic, 1993, pp. 341--369. MR**95j:46074****[G1]**S. Goto,*Orbifold construction for non-AFD subfactors*, Int. J. of Math**5**(1994), 725--746. MR**95h:46094****[G2]**S. Goto,*Symmetric flat connections triviality of Loi's invariant and orbifold subfactors*, to appear in Publ. RIMS Kyoto Univ.**[I]**M. Izumi,*Application of fusion rules to classification of subfactors*, Publ. RIMS Kyoto Univ**27**(1991), 953--994. MR**93b:46121****[Ka1]**Y. Kawahigashi,*Automorphism commuting with a conditional expectation onto a subfactor with finite index*, J. Operator theory**28**(1992), 127--145. MR**95b:46086****[Ka2]**Y. Kawahigashi,*On flatness of Ocneanu's connections on the Dynkin diagramsand classification of subfactors*, J. Funct. Anal.**127**(1995), 63--107. MR**95j:46075****[Ka3]**Y. Kawahigashi,*Centrally trivial automorphisms and an analogue of Connes's for subfactors*, Duke Math. J.**71**(1993), 93--118. MR**94k:46131****[Ko]**H. Kosaki,*Automorphisms in the irreducible decompositions of sectors*, ``Quantum and non-commutative analysis", Kluwer Academic, 1993, pp. 305--316. MR**95e:46073****[L]**P. H. Loi,*On automorphisms of subfactors*, preprint, 1990.**[O1]**A. Ocneanu,*Quantized group string algebras and Galois theory for algebras*, ``Operator algebras and applications, Vol. 2 (Warwick, 1987)'', London Math. Soc. Lect. Note Series Vol. 136, Cambridge University Press, 1988, pp. 119--172. MR**91k:46068****[P1]**S. Popa,*On the classification of actions of amenable groups on subfactors*, C. R. Acad. Sc. Paris.**315**(1992), 295--299. MR**93i:46110****[P2]**S. Popa,*Classification of actions of discrete amenable groups on amenable subfactors of type II*, preprint, 1992.**[P3]**S. Popa,*Classification of amenable subfactors of type II Acta Math.*, vol. 172, 1994, pp. 352--445. MR**95f:46105****[W]**H. Wenzl,*Hecke algebras of type and subfactors*, Invent. Math.**92**(1988), 345--383. MR**90b:46118****[X1]**F. Xu,*Orbifold construction in subfactors*, Comm. Math. Phys.**166**(1994), 237--254. CMP**95:06****[X2]**F. Xu,*The flat part of non-flat orbifolds*, 1993, to appear in Pac. J. of Math.**[Y1]**S. Yamagami,*A note on Ocneanu's approach to Jones' index theory*, Internat. J. of Math.**4**(1993), 859--871. MR**95f:46114****[Y2]**S. Yamagami,*Modular theory for bimodules*, J. Funct. Anal.**125**(1994), 327--357. CMP**95:02**

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

Retrieve articles in all journals with MSC (1991): 46L37

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

DOI:
https://doi.org/10.1090/S0002-9939-96-03443-0

Keywords:
Approximately inner automorphism,
centrally trivial automorphism,
Loi's invariant,
non-strongly outer automorphism,
orbifold construction,
quantum $SU(n)_{k}$ subfactor,
relative Connes invariant

Received by editor(s):
March 9, 1995

Received by editor(s) in revised form:
May 8, 1995

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1996
American Mathematical Society