On the existence of central sequences in subfactors
Author:
Dietmar H. Bisch
Journal:
Trans. Amer. Math. Soc. 321 (1990), 117-128
MSC:
Primary 46L35
DOI:
https://doi.org/10.1090/S0002-9947-1990-1005075-5
MathSciNet review:
1005075
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Abstract | References | Similar Articles | Additional Information
Abstract: We prove a relative version of [Co1, Theorem 2.1] for a pair of type -factors
. This gives a list of necessary and sufficient conditions for the existence of nontrivial central sequences of
contained in the subfactor
. As an immediate application we obtain a result by Bédos [Be, Theorem A], showing that if
has property
and
is an amenable group acting freely on
via some action
, then the crossed product
has property
. We also include a proof of a relative Mc Duff-type theorem (see [McD, Theorems
,
and
]), which gives necessary and sufficient conditions implying that the pair
is stable.
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- [Ta1] M. Takesaki, On the singularity of a positive linear functional, Proc. Japan Acad. 35 (1959), 365-366. MR 0113153 (22:3991)
- [Ta2] -, Theory of operator algebras I, Springer, Berlin, Heidelberg, and New York, 1979. MR 548728 (81e:46038)
- [Be]
E. Bédos, On actions of amenable groups on
-factors, Preprint 1988.
- [Co1]
A. Connes, Classification of injective factors, Cases
, Ann. of Math. 104 (1976), 73-115. MR 0454659 (56:12908)
- [Co2] -, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. 8 (1975), 383-420. MR 0394228 (52:15031)
- [CoFW] A. Connes, J. Feldman and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynamical Systems 1 (1981), 431-450. MR 662736 (84h:46090)
- [Jo] V. F. R. Jones, Index of subfactors, Invent. Math. 72 (1983), 1-25. MR 696688 (84d:46097)
- [Oc] A. Ocneanu, Quantized group string algebras and Galois theory for algebras, preprint. MR 996454 (91k:46068)
- [Po1] S. Popa, A short proof of "Injectivity implies Hyperfiniteness" for finite von Neumann-Algebras, J. Operations Theory 16 (1986), 261-272. MR 860346 (87m:46115)
- [Po2] -, Maximal injective subalgebras in factors associated with free groups, Adv. in Math. 50 (1983), 27-48. MR 720738 (85h:46084)
- [Po3] -, Classification of subfactors: reduction to commuting squares, Invent. Math. (to appear). MR 1055708 (91h:46109)
- [McD] D. McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21 (1970), 443-461. MR 0281018 (43:6737)
- [MvN] F. Murray and J. von Neumann, Rings of operators IV, Ann. of Math. 44 (1943), 716-808. MR 0009096 (5:101a)
- [Ta1] M. Takesaki, On the singularity of a positive linear functional, Proc. Japan Acad. 35 (1959), 365-366. MR 0113153 (22:3991)
- [Ta2] -, Theory of operator algebras I, Springer, Berlin, Heidelberg, and New York, 1979. MR 548728 (81e:46038)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1990-1005075-5
Article copyright:
© Copyright 1990
American Mathematical Society