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

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.