Actions of compact Abelian groups on semifinite injective factors. (English) Zbl 0588.46042
The paper presents an important step in the classification of group actions on von Neumann algebras. The main theorem gives a necessary and sufficient condition for two actions of a compact Abelian group on semifinite injective factors to be conjugate. In these conditions certain invariants are exploited which remind and develop Connes’ invariants for actions of \({\mathbb{Z}}\) on the hyperfinite \(II_ 1\) factor. The classification obtained is far from being smooth. One of the ”parameters” is a nonsingular ergodic action of a countable discrete Abelian group (the dual one) defined up to conjugacy. Another invariant is of cohomological character and little is known about the range of its possible values in the general situation, the picture being ”roughly the same as it is in the classification of injective type \(III_ 0\) factors”, and the main result ”should be thought of as a structure theorem rather than a classification in the sense of enumeration of all possible actions”. The classification is shown to be more simple in some particular cases.
In the subsequent paper: [C. Sutherland and M. Takesaki, Publ. Res. Inst. Math. Sci. Kyoto Univ., 21, 1087-1120 (1985)] the case of discrete amenable groups is considered.
In the subsequent paper: [C. Sutherland and M. Takesaki, Publ. Res. Inst. Math. Sci. Kyoto Univ., 21, 1087-1120 (1985)] the case of discrete amenable groups is considered.
Reviewer: A.Lodkin
Keywords:
classification of group actions on von Neumann algebras; semifinite injective factors; Connes’ invariants for actions of \({\mathbb{Z}}\) on the hyperfinite \(II_ 1\) factorReferences:
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