Flows Associated with Ergodic Non-Singular Transformation Groups

Abstract

In this paper we shall give a nice invariant for the weak equivalence of ergodic non-singular transformation groups. It is a one-parameter ergodic non-singular flow associated with an ergodic non-singular transformation group. Since in 1960 an example of an ergodic non-singular transformation without <r-finite invariant measures was given in Ergodic theory [16], the structure and the classification of ergodic non-singular transformations have been studied by many authors ([2], [4], [5], [6], [7], [8] ~[H] and [13]). Among these works, Krieger's weak equivalence theory is fundamental in the classification problem of ergodic nonsingular transformation groups without er-finite invariant measures. This classification is closely connected with the classification of type III factors in the theory of von Neumann algebras ([15]). The Tomita-Takesaki theory of generalized Hilbert algebras ([18]) plays important roles in the analysis of type III factors. Using this theory, A. Connes [3] introduced algebraic invariants — the S-set S(M) and the T-set T(M) — for a factor M and obtained a classification of type III factors. M. Takesaki [19] introduced the dual action of the modular automorphism group and obtained the structure theorem of type III factors. In the classification problem of ergodic non-singular transformation groups G, W. Krieger [10] and the present authors [6] introduced invariants r(G) and T(G) respectively, both of which are