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The Intrinsic Invariant of an Approximately Finite Dimensional Factor and the Cocycle Conjugacy of Discrete Amenable Group Actions
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The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions
Yoshikazu Katayama, Colin E. Sutherland, and Masamichi Takesaki
Abstract.
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
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