On converse to Koopman's Lemma

Koopman's Lemma states that if a flow Tt is measure preserving for a measure μ on a constant energy surface Ω, then the flow generates a one parameter family of unitary operators Ut on L2 (Ω, μ). We show here a converse, namely that under certain (physically motivated) conditions a unitary operator family Ut can be made to generate a corresponding underlying family Tt of point transformations. This result comes out of questions of independent interest in the study of relationships between reversibility and irreversibility, and has application to the foundations of statistical mechanics. In particular, it establishes the principle often used intuitively in chemistry that a forward moving (e.g., Markov) process that loses information cannot be reversed. In a different setting, it provides the answer to a question in the representation theory of isometries on Lp spaces a Banach-Lamperti theorem). These results also allow an interesting reformulation of Ornstein's isomorphism theorem on Bernoulli systems.