Actions of discrete amenable groups and groupoids on von Neumann algebras (Q1820369)

This paper provides a complete set of invariants, up to cocycle conjugacy, for the actions of discrete amenable ergodic groupoids on injective semi-finite factors, and derives a classification of actions of discrete amenable groups on injective semifinite von Neumann algebras as a corollary. A consequence of this classification is the fact that such actions always admit globally invariant Cartan subalgebras. The result itself is a direct extension of those obtained by \textit{V. F. R. Jones} and \textit{M. Takesaki} in Acta Math. 153, 213-258 (1984; Zbl 0588.46042), where actions of abelian groups are considered. The main new features are devices to handle the case where the isotropy groups of the groupoid vary from point to point; these devices are developed in a companion paper of the first author, ''A Borel parametrization of Polish groups'', Res. Inst. Math. Sci. Kyoto, Vol. 21, No.6, 1067-1086 (1985; Zbl 0605.46050).