Dynamical entropy for Markov operators (Q1570896)

The author defines and examines the entropy for the class of Markov operators, an intermediate case between the classical and quantum systems. The author proves that in the case \(Pf(x)= f(Tx)\), where \(T\) is a measure preserving transformation and \(P\) is a Markov operator, the newly defined entropy coincides with the Kolmogorov-Sinai entropy of \(T\). Moreover the author studies a dilation of the Markov operator \(P\) through the Markov chain associated with \(P\). It is shown that the entropy depends on the projections \(\pi_{-\infty}\) and \(\pi_0\) on \(\sigma\)-algebras \({\mathcal A}_{-\infty}\) and \({\mathcal A}_0\) of the Markov chain.