A characterization of normal extensions for subfactors

Teruya, Tamotsu

doi:10.1090/S0002-9939-1994-1207542-7

A characterization of normal extensions for subfactors
HTML articles powered by AMS MathViewer

by Tamotsu Teruya PDF

Proc. Amer. Math. Soc. 120 (1994), 781-783 Request permission

Abstract:

Let $N \subset M \subset L$ be a tower of factors. If $L$ is a crossed product $N{ \rtimes _\alpha }G$ of $N$ by an outer action $\alpha$ of a finite group $G$ on $N$ then it is well known that there exists a subgroup $H$ of $G$ such that $M = N{ \rtimes _{{\alpha _{{|_H}}}}}H$. We prove in this paper that $H$ is a normal subgroup of $G$ if and only if there exist a finite group $F$ and an outer action $\beta$ of $F$ on $M$ such that $L = M{ \rtimes _\beta }F$.

References

Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799, DOI 10.1007/978-1-4613-9641-3
Vaughan F. R. Jones, Actions of finite groups on the hyperfinite type $\textrm {II}_{1}$ factor, Mem. Amer. Math. Soc. 28 (1980), no. 237, v+70. MR 587749, DOI 10.1090/memo/0237
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127
Hideki Kosaki, Characterization of crossed product (properly infinite case), Pacific J. Math. 137 (1989), no. 1, 159–167. MR 983334
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
Masahiro Nakamura and Zirô Takeda, A Galois theory for finite factors, Proc. Japan Acad. 36 (1960), 258–260. MR 123925

Quantum symmetry, differential geometry of finite graphs and classification of subfactors

Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811
Colin E. Sutherland, Cohomology and extensions of von Neumann algebras. I, II, Publ. Res. Inst. Math. Sci. 16 (1980), no. 1, 105–133, 135–174. MR 574031, DOI 10.2977/prims/1195187501
Zirô Takeda, On the normal basis theorem of the Galois theory for finite factors, Proc. Japan Acad. 37 (1961), 144–148. MR 130874
Yasuo Watatani, Index for $C^*$-subalgebras, Mem. Amer. Math. Soc. 83 (1990), no. 424, vi+117. MR 996807, DOI 10.1090/memo/0424