Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups
type: Item
instance of: scholarly article
title: Groups of measure space transformations and invariants of outer conjugation for automorphisms from normalizers of type III full groups
zbMATH Open document ID: 0557.28018
DOI: 10.1016/0022-1236(85)90044-8
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 3887988
zbMATH Keywords: type of an automorphism
zbMATH Keywords: amenability
zbMATH Keywords: necessary and sufficient conditions of outer conjugation of
zbMATH Keywords: automorphisms
zbMATH Keywords: normalizer of an approximated type III automorphism group
zbMATH Keywords: normalizer
zbMATH Keywords: centralizer
zbMATH Keywords: surjective homomorphism
zbMATH Keywords: standard representation
zbMATH Keywords: outer period
zbMATH Keywords: actions of countable amenable groups
zbMATH Keywords: necessary and sufficient conditions of outer conjugation of automorphisms
author: S. Bezuglyi
author: V. Ya. Golodets
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90044-8
OpenAlex ID: W1963624224
cites work: Outer conjugacy classes of automorphisms of factors
cites work: The normalizer group of an ergodic automorphism of type III and the commutant of an ergodic flow
cites work: Measure space automorphisms, the normalizers of their full groups, and approximate finiteness
cites work: The flow of weights on factors of type III
cites work: On ergodic flows and the isomorphism of factors
cites work: On Groups of Measure Preserving Transformations. I
cites work: Q5615484
cites work: Flows associated with ergodic non-singular transformation groups
cites work: Q3937606
cites work: Q5790284
cites work: Q5541832
cites work: Q3926822
cites work: Actions of compact Abelian groups on semifinite injective factors
cites work: Orbit structure and countable sections for actions of continuous groups
description: scientific article; zbMATH DE number 3887988