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We announce in this article that i) to each approximately finite dimensional factor $\r$ of any type there corresponds canonically a group cohomological invariant, to be called the intrinsic invariant of $\r$ and denoted $\Theta(\r)$, on which $\Aut(\r)$ acts canonically; ii) when a group $G$ acts on $\r$ via $\a: G \mapsto \Aut(\r)$, the pull back of Orb($\Theta (\r)$), the orbit of $\Theta(\r)$ under $\Aut(\r)$,by $\a$ is a cocycle conjugacy invariant of $\a$; iii) if $G$ is a discrete countable amenable group, then the pair of the module, mod($\a$), and the above pull back is a complete invariant for the cocycle conjugacy class of $\a$. This result settles the open problem of the general cocycle conjugacy classification of discrete amenable group actions on an AFD factor of type ${\hbox{\uppercase\expandafter {\romannumeral3}}}_1$, and unifies known results for other types.

Copyright American Mathematical Society 1995

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Article Info

• ERA Amer. Math. Soc. 01 (1995), pp. 43-47
• Publisher Identifier: S 1079-6762(95)01006-1
• 1991 Mathematics Subject Classification. 46L40 .
• Received by the editors May 17, 1995
• Comments (When Available)

Yoshikazu Katayama

Department of Mathematics, Osaka Kyoiku University, Osaka, Japan.

E-mail address: F61021@sinet.adjp

Colin E. Sutherland

Department of Mathematics, University of New South Wales, Kensington, NSW, Australia.

E-mail address: colins@solution.maths.unsw.edu.au

Masamichi Takesaki

Department of Mathematics, University of California, Los Angeles, Califnornia 90024-1555.

E-mail address: mt@math.ucla.edu

This research is supported in part by NSF Grant DMS92-06984 and DMS95-00882, and also supported by the Australian Research Council Grant