Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups
The paper attemps to find the necessary and sufficient conditions of outer conjugation of automorphisms from the normalizer of an approximated type III automorphism group. Let T be an automorphism of type \(III_ 0\), [T] the full group of automorphisms generated by T, N[T] the normalizer of the full group [T], \(\{W_ T(\cdot)\}\) the flow of automorphisms associated with T, \(C\{W_ T(\cdot)\}\) the centralizer of the flow \(\{W_ T(\cdot)\}\). The main results are: (1) the mapping of \(\alpha \to mod \alpha: N[T]\to C\{W_ T(\cdot)\}\) is a surjective homomorphism; (2) a standard representation is found for any automorphism from N[T]; (3) automorphisms \(\alpha_ 1,\alpha_ 2\in N[T]\) are outer conjugate if and only if \(p(\alpha_ 1)=p(\alpha_ 2)\), mod \(\alpha\) \({}_ 1=\gamma mod \alpha_ 2\gamma^{-1}\), where p(\(\cdot)\) is the outer period and \(\gamma \in C\{W_ T\}.\) Similar statements are proved for the case that T is type \(III_{\lambda}\) \((0<\lambda <1)\) and type \(III_ 1.\) At present the authors have obtained more general results on outer conjugation. Namely, the necessary and sufficient conditions have been found of outer conjugation for actions of countable amenable groups of automorphisms from the normalizer of the full group of automorphisms [T]. The cases of type II and type III are considered.
necessary and sufficient conditions of outer conjugation of
normalizer of an approximated type III automorphism group
necessary and sufficient conditions of outer conjugation of automorphisms
scientific article; zbMATH DE number 3887988