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Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups
Journal article   Open access   Peer reviewed

Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups

S. I Bezuglyi and V. Ya Golodets
Journal of Functional Analysis, Vol.60(3), pp.341-369
1985
DOI: 10.1016/0022-1236(85)90044-8
url
https://doi.org/10.1016/0022-1236(85)90044-8View
Published (Version of record) Open Access

Abstract

The necessary and sufficient conditions of outer conjugation for automorphisms from the normalizer of approximated III type groups are found. Let T be an automorphism of a Lebesgue space ( X , μ ) of the III 0 type, [ T ] the full group generated by T , N [ T ] its normalizer, { W t ( T )} the flow associated with T and α → mod α the homomorphism from N [ T ] to C { W } the centralizer of the associated flow. The following results are obtained: ∀α ∈ C{W} ∃ \ ̂ ga ∈ N[T] such that mod \ ̂ ga = α; automorphisms α 1 , and α 2 from N [ T ] are outer conjugate if and only if p ( α 1 ) = p ( α 2 ), mod α 1 = γ mod α 2 γ −1 , where γ ∈ C { W } and p (·) is the outer period; the canonical form of the elements from N [ T ] is found. The case is also considered where T is types III λ (0 < λ < 1) and III 1 .

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