The Topological Structure of Polish Groups and Groupoids of Measure Space Transformations

Abstract

It is proved that the groupoid of nonsingular partial isomorphisms of a Lebesgue space (X,μ) is weakly contractible in a “strong” sense: we present a contraction path which preserves invariant the subgroupoid of μ-preserving partial isomorphisms as well as the group of nonsingular transformations of X. Moreover, let R be an ergodic measured discrete equivalence relation on X. The full group [R] endowed with the uniform topology is shown to be contractible. For an approximately finite R of type II or IIIλ, 0 ≤ λ < 1, the normalizer N [R] of R furnished with the natural Polish topology is established to be homotopically equivalent to the centralizer of the associated Poincare flow. These are the measure theoretical analogues of the resent results of S. Popa and M. Takesaki on the topological structure of the unitary and the automorphism group of a factor. The topological properties of automorphism groups of a Lebesgue space (X,μ) have been studied since 1944, when P. Halmos [Ha] introduced two metrizable topologies on the group Aut0(X,μ) of μ-preserving transformations: the weak dw and the uniform du. He proved that (Aut0(X,μ), dw) is a Polish group. S. Harada [Har] showed that it is simply connected and arcwise connected. His result was later considerably refined by M. Keane [K] who proved the contractibility of Aut0(X,μ) both in du and dw (see also [D, N]). A. Ionescu Tulsea [IT] and R. V. Chacon and N. A. Friedman [ChF] extended the weak and the uniform topology to the group Aut (X,μ) of nonsingular transformations of (X,μ) and generalized the results obtained in [Ha]. However, the homotopical properties of this group have not been studied so far. For a detailed exposition of the productive interplay between ergodic theory and operator algebras we refer to [M, C, S2]. The present work also was stimulated by a paper [PT] being pertained to operator algebras. In particular, given a countable group Γ ⊂ Aut (X,μ), then one can consider the full group [Γ] and its normalizer N [Γ] in Aut (X,μ) which are the measure theoretical analogues of the unitary group U(M) and the automorphism group Aut (M) of a von Neumann algebra M . Both groups, [Γ] andN [Γ], are Polish: the first with respect to du, the second with respect to some metric d defined by T. Hamachi and M. Osikawa [HO]. Further topological 1991 Mathematics Subject Classification. Primary 55P10, 22A05, 22A22, 28D15; Secondary 46L55.