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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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