Actions of Discrete Amenable Groups and Groupoids on von Neumann Algebras

Abstract

The purpose of this paper is to provide a complete set of invariants, up to cocycle conjugacy, of the possible actions, a, of a discrete amenable group G on a semifinite injective von Neumann algebra Jt. In the case where Ji = L°° (X', ft), this amounts to giving invariants, up to conjugacy, for the non-singular actions of G on a Lebesgue space (X,fjt), a problem which is unsolved even for G = Z', in the general case, the action of G on the center of JK itself appears as part of the invariant. If Ji is a factor and G = ZP or Z, the problem was solve by fundamental work of A. Gonnes in [2, 4]; refining Connes' techniques, V. Jones, [9], resolved the case where J£ is a factor and G is finite, and A. Ocneanu, [14], resolved the case where Ji is a factor and G is amenable. Also, in [10], Jones and Takesaki gave a complete set of invariants for the case where Jt is no longer a factor, but G is abelian. The main theorem of the present paper subsumes all these results, and depends crucially on the results of Ocneanu in [14], and the techniques developed by Jones and Takesaki in [10] to handle the non-factor case. For technical simplicity, we treat the case where the restriction of a to the center ^(^) is ergodic (i.e. a is centrally ergodic) ; we let