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A characterization of a coaction reduced to that of a closed subgroup. (English) Zbl 1156.46043
Summary: It is shown that, for any coaction a of a locally compact group $$K$$ on a properly infinite von Neumann algebra $$A$$ and a closed subgroup $$H$$ of $$K$$, $$\alpha$$ is cocycle conjugate to a coaction which comes from a coaction of $$H$$ if and only if the dual action $$\widehat\alpha$$ is induced by an action of $$H$$. We also include applications of the result concerning almost periodic coactions and the ranges of 1-cocycles on measured equivalence relations.
##### MSC:
 46L55 Noncommutative dynamical systems
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##### References:
 [1] H. Aoi and T. Yamanouchi, A characterization of coactions whose fixed-point algebras contain special maximal abelian $$\ast$$-subalgebras, Ergod. Th. & Dynam. Sys., 26 (2006), 1673-1706. · Zbl 1135.46038 [2] H. Aoi and T. Yamanouchi, On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations-subrelations, J. Funct. Anal., 240 (2006), 297-333. · Zbl 1122.28012 [3] K. Dykema, Crossed product decompositions of a purely infinite von Neumann algebra with faithful, almost periodic weight, Indiana Univ. Math. J., 44 (1995), no. 2, 433-450. · Zbl 0841.46048 [4] G. B. Folland, A Course in Abstract Harmonic Analysis , RC Press, Inc., 1995. · Zbl 0857.43001 [5] P. Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc., 242 (1978), 35-72. · Zbl 0356.46055 [6] M. Izumi, Canonical extension of endomorphisms of type III factors, Amer. J. Math., 125 (2003), no. 1, 1-56. · Zbl 1037.46054 [7] M. B. Landstad, Twisted dual-group algebras: Equivalent reformations of $$C_0(G)$$, J. Funct. Anal., 132 (1995), 43-85. · Zbl 0839.22003 [8] Y. Nakagami, Dual action on a von Neumann algebra and Takesaki’s duality for a locally compact group, Publ. RIMS, Kyoto Univ., 12 (1977), 727-775. · Zbl 0363.46062 [9] Y. Nakagami and M. Takesaki, Duality for crossed products of von Neumann algebras , Lecture Notes in Math. 731 , Springer-Verlag, 1979. · Zbl 0423.46051 [10] M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math., 131 (1973), 249-310. · Zbl 0268.46058 [11] T. Yamanouchi, One-cocycles on smooth flows of weights and extended modular coactions, Ergod. Th. & Dynam. Sys., 27 (2007), 285-318. · Zbl 1152.46057
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