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Ergodic transformations conjugate to their inverses by involutions. (English) Zbl 0846.28008
The authors consider invertible ergodic transformations \(T\) of a Borel probability space such that \(T\) and \(T^{- 1}\) are isomorphic (that is there exists an invertible measure-preserving transformation \(S\) such that \(TS= ST^{- 1}\)). If \(T\) has discrete spectrum, then it is known that \(S^2= I\). The authors extend this to the case where \(T\) has simple spectrum. They also show that if instead \(T\) is such that the commutant of \(T\) (the set of measure-preserving transformations which commute with \(T\)) is the weak closure of \(\{T^n: n\in \mathbb{Z}\}\) and \(T\) is isomorphic to its inverse, then the conjugating isomorphism \(S\) satisfies \(S^4= I\).
The authors give a number of examples including one which has the above ‘rank closure’ property for which \(S^2\neq I\). The techniques used are those of unitary operators on Hilbert space.
Reviewer: A.Quas (Cambridge)

MSC:
28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
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