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ISSN 1079-6762



The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions

Authors: Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 43-47
MSC (1991): Primary 46L40
MathSciNet review: 1336699
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Abstract: We announce in this article that i) to each approximately finite dimensional factor $\mathcal {R}$ of any type there corresponds canonically a group cohomological invariant, to be called the intrinsic invariant of $\mathcal {R}$ and denoted $\Theta (\mathcal {R})$, on which $\operatorname {Aut}(\mathcal {R})$ acts canonically; ii) when a group $G$ acts on $\mathcal {R}$ via $\alpha : G \mapsto \operatorname {Aut}(\mathcal {R})$, the pull back of Orb($\Theta (\mathcal {R})$), the orbit of $\Theta (\mathcal {R})$ under $\operatorname {Aut}(\mathcal {R})$, by $\alpha$ is a cocycle conjugacy invariant of $\alpha$; iii) if $G$ is a discrete countable amenable group, then the pair of the module, mod($\alpha$), and the above pull back is a complete invariant for the cocycle conjugacy class of $\alpha$. This result settles the open problem of the general cocycle conjugacy classification of discrete amenable group actions on an AFD factor of type $\mathrm {III}_1$, and unifies known results for other types.

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Additional Information

Yoshikazu Katayama
Affiliation: address Yoshikazu Katayama, Department of Mathematics, Osaka Kyoiku University, Osaka, Japan.
Email: F61021@sinet.adjp

Colin E. Sutherland
Affiliation: address Colin E. Sutherland, Department of Mathematics, University of New South Wales, Kensington, NSW, Australia.

Masamichi Takesaki
Affiliation: Department of Mathematics, University of California, Los Angeles, Califnornia 90024-1555.

Received by editor(s): May 17, 1995
Additional Notes: This research is supported in part by NSF Grant DMS92-06984 and DMS95-00882, and also supported by the Australian Research Council Grant
Article copyright: © Copyright 1995 American Mathematical Society