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

Katayama, Yoshikazu¹ ; Sutherland, Colin² ; Takesaki, Masamichi³

¹ address Yoshikazu Katayama, Department of Mathematics, Osaka Kyoiku University, Osaka, Japan.
² address Colin E. Sutherland, Department of Mathematics, University of New South Wales, Kensington, NSW, Australia.
³ Department of Mathematics, University of California, Los Angeles, Califnornia 90024-1555.

Electronic research announcements of the American Mathematical Society, Tome 01 (1995) no. 1, pp. 43-47.

Résumé

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.

Détail

DOI : 10.1090/S1079-6762-95-01006-7

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