Decreasing nets of \(\sigma\)-algebras and their applications to ergodic theory (Q1177411)

This paper is based on the author's extension of the Rohlin-Sinai theorem on the existence of perfect \(\sigma\)-algebras to factors of \(\mathbb{Z}^ d\)- actions. Moreover it uses (and gives new proofs of) the fact that the order of taking suprema and countable intersections of \(\sigma\)-algebras can be exchanged under suitable assumptions of independence. The main result states: For a \(\mathbb{Z}^ d\)-action \(\Phi\) and a factor \(\mathcal H\) of \(\Phi\) the following are equivalent: a) \(\mathcal H\) is entropy maximal for \(\Phi\), i.e., for every factor \({\mathcal C}\supset{\mathcal H}\) with entropy \(h(\Phi/{\mathcal C}/{\mathcal H})=0\) one has \({\mathcal C}={\mathcal H}\). b) \(\Phi\) is a relative \(K\)-action for \(\mathcal H\), i.e., there is an exhaustive sub-\(\sigma\)-algebra \({\mathcal A}\supset{\mathcal H}\) with \({\mathcal H}=\bigcap_{g\in\mathbb{Z}^ d}\Phi^ g{\mathcal A}={\mathcal H}\). Also a Pollit formula for \(\mathbb{Z}^ d\)-actions is given.