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A classification of flows on AFD factors with faithful Connes-Takesaki modules


Author: Koichi Shimada
Journal: Trans. Amer. Math. Soc. 368 (2016), 4497-4523
MSC (2010): Primary 46L10
DOI: https://doi.org/10.1090/tran/6471
Published electronically: October 14, 2015
MathSciNet review: 3453378
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Abstract: We completely classify flows on approximately finite dimensional (AFD) factors with faithful Connes-Takesaki modules up to cocycle conjugacy. This is a generalization of the uniqueness of the trace-scaling flow on the AFD factor of type $ \mathrm {II}_\infty $, which is equivalent to the uniqueness of the AFD factor of type $ \mathrm {III}_1$. In order to achieve this, we show that a flow on any AFD factor with faithful Connes-Takesaki module has the Rohlin property, which is a kind of outerness for flows introduced by Kishimoto and Kawamuro.


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  • [1] A. Connes, Almost periodic states and factors of type $ {\rm III}_{1}$, J. Functional Analysis 16 (1974), 415-445. MR 0358374 (50 #10840)
  • [2] A. Connes, Factors of type $ {\rm III}_1$, property $ L_\lambda '$ and closure of inner automorphisms, J. Operator Theory 14 (1985), no. 1, 189-211. MR 789385 (88b:46088)
  • [3] Alain Connes, Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 383-419. MR 0394228 (52 #15031)
  • [4] Alain Connes and Masamichi Takesaki, The flow of weights on factors of type $ {\rm III}$, Tôhoku Math. J. (2) 29 (1977), no. 4, 473-575. MR 480760 (82a:46069a), https://doi.org/10.2748/tmj/1178240493
  • [5] J. Feldman, Changing orbit equivalences of $ {\bf R}^d$ actions, $ d\geq 2$, to be $ {\mathcal {C}}^\infty $ on orbits, Internat. J. Math. 2 (1991), no. 4, 409-427. MR 1113569 (93e:58108a), https://doi.org/10.1142/S0129167X91000235
  • [6] J. Feldman, $ \mathrm {C}^\infty $ orbit equivalence of flows, Xeroxed lecture notes, privately circulated.
  • [7] Jacob Feldman, Peter Hahn, and Calvin C. Moore, Orbit structure and countable sections for actions of continuous groups, Adv. in Math. 28 (1978), no. 3, 186-230. MR 0492061 (58 #11217)
  • [8] J. Feldman and D. A. Lind, Hyperfiniteness and the Halmos-Rohlin theorem for nonsingular Abelian actions, Proc. Amer. Math. Soc. 55 (1976), no. 2, 339-344. MR 0409764 (53 #13516)
  • [9] Masaki Izumi, Canonical extension of endomorphisms of type III factors, Amer. J. Math. 125 (2003), no. 1, 1-56. MR 1953517 (2003k:46090)
  • [10] Vaughan F. R. Jones, Actions of finite groups on the hyperfinite type $ {\rm II}_{1}$factor, Mem. Amer. Math. Soc. 28 (1980), no. 237, v+70. MR 587749 (81m:46094), https://doi.org/10.1090/memo/0237
  • [11] Yoshikazu Katayama, Colin E. Sutherland, and Masamichi Takesaki, The characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors, Invent. Math. 132 (1998), no. 2, 331-380. MR 1621416 (99f:46096), https://doi.org/10.1007/s002220050226
  • [12] Yasuyuki Kawahigashi, One-parameter automorphism groups of the injective $ {\rm II}_1$ factor arising from the irrational rotation $ C^*$-algebra, Amer. J. Math. 112 (1990), no. 4, 499-523. MR 1064989 (91m:46106), https://doi.org/10.2307/2374868
  • [13] Yasuyuki Kawahigashi, One-parameter automorphism groups of the injective factor of type $ {\rm II}_1$ with Connes spectrum zero, Canad. J. Math. 43 (1991), no. 1, 108-118. MR 1108916 (92f:46074), https://doi.org/10.4153/CJM-1991-007-9
  • [14] Y. Kawahigashi, C. E. Sutherland, and M. Takesaki, The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions, Acta Math. 169 (1992), no. 1-2, 105-130. MR 1179014 (93j:46068), https://doi.org/10.1007/BF02392758
  • [15] Keiko Kawamuro, A Rohlin property for one-parameter automorphism groups of the hyperfinite $ \rm II_1$ factor, Publ. Res. Inst. Math. Sci. 36 (2000), no. 5, 641-657. MR 1798489 (2002a:46093), https://doi.org/10.2977/prims/1195142813
  • [16] A. Kishimoto, A Rohlin property for one-parameter automorphism groups, Comm. Math. Phys. 179 (1996), no. 3, 599-622. MR 1400754 (97e:46092)
  • [17] Izumi Kubo, Quasi-flows, Nagoya Math. J. 35 (1969), 1-30. MR 0247032 (40 #301)
  • [18] Uffe Haagerup, Connes' bicentralizer problem and uniqueness of the injective factor of type $ {\rm III}_1$, Acta Math. 158 (1987), no. 1-2, 95-148. MR 880070 (88f:46117), https://doi.org/10.1007/BF02392257
  • [19] Uffe Haagerup and Erling Størmer, Equivalence of normal states on von Neumann algebras and the flow of weights, Adv. Math. 83 (1990), no. 2, 180-262. MR 1074023 (92d:46150), https://doi.org/10.1016/0001-8708(90)90078-2
  • [20] Uffe Haagerup and Erling Størmer, Pointwise inner automorphisms of von Neumann algebras, J. Funct. Anal. 92 (1990), no. 1, 177-201. With an appendix by Colin Sutherland. MR 1064693 (92c:46067), https://doi.org/10.1016/0022-1236(90)90074-U
  • [21] Christopher Lance, Direct integrals of left Hilbert algebras, Math. Ann. 216 (1975), 11-28. MR 0372626 (51 #8833)
  • [22] D. A. Lind, Locally compact measure preserving flows, Advances in Math. 15 (1975), 175-193. MR 0382595 (52 #3477)
  • [23] Toshihiko Masuda, Unified approach to the classification of actions of discrete amenable groups on injective factors, J. Reine Angew. Math. 683 (2013), 1-47. MR 3181546, https://doi.org/10.1515/crelle-2011-0011
  • [24] T. Masuda and R. Tomatsu, Classification of actions of discrete Kac algebras on injective factors, preprint (2013), arXiv:1306.5046.
  • [25] T. Masuda and R. Tomatsu, Rohlin flows on von Neumann algebras, preprint (2012), arXiv:1206.0955.
  • [26] Adrian Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, vol. 1138, Springer-Verlag, Berlin, 1985. MR 807949 (87e:46091)
  • [27] Koichi Shimada, Rohlin flows on amalgamated free product factors, Int. Math. Res. Not. IMRN 3 (2015), 773-786. MR 3340336
  • [28] Colin E. Sutherland and Masamichi Takesaki, Actions of discrete amenable groups on injective factors of type $ {\rm III}_\lambda ,\;\lambda \neq 1$, Pacific J. Math. 137 (1989), no. 2, 405-444. MR 990219 (90k:46142)
  • [29] Colin E. Sutherland and Masamichi Takesaki, Right inverse of the module of approximately finite-dimensional factors of type III and approximately finite ergodic principal measured groupoids, Operator algebras and their applications, II (Waterloo, ON, 1994/1995), Fields Inst. Commun., vol. 20, Amer. Math. Soc., Providence, RI, 1998, pp. 149-159. MR 1643188 (99k:46108)
  • [30] M. Takesaki, Theory of operator algebras. I, reprint of the first (1979) edition, Encyclopaedia of Mathematical Sciences, vol. 124. Operator Algebras and Non-commutative Geometry, 5, Springer-Verlag, Berlin, 2002. MR 1873025 (2002m:46083)
  • [31] M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Operator Algebras and Non-commutative Geometry, 6, Springer-Verlag, Berlin, 2003. MR 1943006 (2004g:46079)
  • [32] Takehiko Yamanouchi, One-cocycles on smooth flows of weights and extended modular coactions, Ergodic Theory Dynam. Systems 27 (2007), no. 1, 285-318. MR 2297097 (2008i:46055), https://doi.org/10.1017/S0143385706000551

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Additional Information

Koichi Shimada
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan
Email: shimada@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/tran/6471
Received by editor(s): August 9, 2013
Received by editor(s) in revised form: February 23, 2014, April 11, 2014, and May 9, 2014
Published electronically: October 14, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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