The Inverse Stable Range Functor

Abstract

We give an inverse construction of the stable range for general flows which may or may not admit an invariant measure. The inverse map is then shown to be a right inverse functor of the stable range functor. The Krieger theorem asserts that in the type IIIo case, there is a one-toone correspondence between orbit equivalent classes of ergodic transformations, algebraic isomorphism classes of Krieger factors, and conjugate isomorphism classes of nontransitive ergodic flows. This correspondence is established via explicit maps between the objects, namely the stable range map, the crossed product, and the flow of weights. In this paper we are concerned with inverting the stable range map, i.e., given a flow on a standard measure space, we will explicitly construct a transformation with the flow as its stable range. Since orbit equivalent transformations give rise to conjugate flows under the stable range map, the choice of the transformation is not unique. However, by using some kind of skew product with the odometer, it is possible to choose a natural one which satisfies the requirement. This construction provides the vital missing arrow which points in the opposite direction of the other maps. For the case where the flow admits an invariant measure, such a transformation can be found in [4] or [5], but for a general flow the problem is more difficult. In [3] Hamachi showed that for a given ergodic flow, one can construct an ergodic action of Z x (Z + aZ) on a certain standard measure space whose stable range is isomorphic to the flow. The result of Connes, Feldman, and Weiss guarantees that the equivalence relation generated by this action is generated by a single transformation T, but it is far from clear how to construct such a transformation explicitly. Noting that the stable range map is in fact a functor from the category of transformations with orbit transporting isomorphisms to the category of flows with conjugations, we see that the construction below also extends to the construction of a right inverse functor to the stable range functor. This fact establishes that the mod homomorphism from the automorphism group of a measured equivalence relation to the automorphism group of its associated flow is split (this also follows from Hamachi's construction [3, p. 399]). Our method also yields a transparent proof of surjectivity of mod, a fact noted by both Hamachi and Golodets [1]. Received by the editors June 8, 1990 and, in revised form, November 7, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28D99. ? 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page