On finiteness of the set of intermediate subfactors

Khoshkam, M.; Mashood, B.

doi:10.1090/S0002-9939-04-07448-9

On finiteness of the set of intermediate subfactors
HTML articles powered by AMS MathViewer

by M. Khoshkam and B. Mashood PDF

Proc. Amer. Math. Soc. 132 (2004), 2939-2944 Request permission

Abstract:

For type $II_1$ factors $N\subset L$ with $[L:N]<\infty$, we show that the sets $\mathcal {L}_1\!=\!\{M\!\in \! \mathcal {L}(N\!\subset \! L)\colon N’\cap L\! \subset \! M\}$ and $\mathcal {L}_2\!=\!\{M\!\in \! \mathcal {L}(N\!\subset \! L)\colon N’\cap L \!=\!M’\cap L\}$ are finite. Moreover, $\mathcal {L}(N\subset L)$, the set of intermediate subfactors, is finite if and only if it is equal to $\mathcal {L}_1\cup \mathcal {L}_2$. If $N$ is an irreducible subfactor, then we recover a result of Y. Watatani.

References

Dietmar Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), no. 2, 201–216. MR 1262294, DOI 10.2140/pjm.1994.163.201
Erik Christensen, Subalgebras of a finite algebra, Math. Ann. 243 (1979), no. 1, 17–29. MR 543091, DOI 10.1007/BF01420203
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127
M. Khoshkam and B. Mashood, On the relative commutants of subfactors, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2725–2732. MR 1451814, DOI 10.1090/S0002-9939-98-04345-7
B. Mashood, Ph.D. thesis, University of Saskatchewan.
Bahman Mashhood and Keith F. Taylor, On continuity of the index of subfactors of a finite factor, J. Funct. Anal. 76 (1988), no. 1, 56–66. MR 923044, DOI 10.1016/0022-1236(88)90048-1
Tamotsu Teruya and Yasuo Watatani, Lattices of intermediate subfactors for type III factors, Arch. Math. (Basel) 68 (1997), no. 6, 454–463. MR 1444657, DOI 10.1007/s000130050078
Yasuo Watatani, Lattices of intermediate subfactors, J. Funct. Anal. 140 (1996), no. 2, 312–334. MR 1409040, DOI 10.1006/jfan.1996.0110