Publications of the Research Institute for Mathematical Sciences


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Volume 31, Issue 5, 1995, pp. 913–940
DOI: 10.2977/prims/1195163723

Published online: 1995-10-31

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

Alexandre I Danilenko[1]

(1) Kharkov State University, Ukraine

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 PoincarĂ© 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.

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Danilenko Alexandre: The Topological Structure of Polish Groups and Groupoids of Measure Space Transformations. Publ. Res. Inst. Math. Sci. 31 (1995), 913-940. doi: 10.2977/prims/1195163723