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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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