On the existence of central sequences in subfactors
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- by Dietmar H. Bisch PDF
- Trans. Amer. Math. Soc. 321 (1990), 117-128 Request permission
Abstract:
We prove a relative version of [Co1, Theorem 2.1] for a pair of type ${\text {I}}{{\text {I}}_1}$-factors $N \subset M$. This gives a list of necessary and sufficient conditions for the existence of nontrivial central sequences of $M$ contained in the subfactor $N$. As an immediate application we obtain a result by Bédos [Be, Theorem A], showing that if $N$ has property $\Gamma$ and $G$ is an amenable group acting freely on $N$ via some action $\sigma$, then the crossed product $N{ \times _\sigma }G$ has property $\Gamma$. We also include a proof of a relative Mc Duff-type theorem (see [McD, Theorems $1$, $2$ and $3$]), which gives necessary and sufficient conditions implying that the pair $N \subset M$ is stable.References
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- Sorin Popa, A short proof of “injectivity implies hyperfiniteness” for finite von Neumann algebras, J. Operator Theory 16 (1986), no. 2, 261–272. MR 860346
- Sorin Popa, Maximal injective subalgebras in factors associated with free groups, Adv. in Math. 50 (1983), no. 1, 27–48. MR 720738, DOI 10.1016/0001-8708(83)90033-6
- S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), no. 1, 19–43. MR 1055708, DOI 10.1007/BF01231494
- Dusa McDuff, Central sequences and the hyperfinite factor, Proc. London Math. Soc. (3) 21 (1970), 443–461. MR 281018, DOI 10.1112/plms/s3-21.3.443
- F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716–808. MR 9096, DOI 10.2307/1969107
- Masamichi Takesaki, On the singularity of a positive linear functional on operator algebra, Proc. Japan Acad. 35 (1959), 365–366. MR 113153
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
Additional Information
- © Copyright 1990 American Mathematical Society
- 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