On the entropy for actions of some class of groups and its calculation

Анотація

The dynamical entropy for actions of $\bigoplus Z_k, k=2,3,\ldots,$ on $C^*{\text -}$algebras which is studied in this work, is a generalization of Connes-Narnhofer-Thirring entropy for actions of the torsion groups on $C^*{\text -}$algebras. The properties of such entropy are investigated and a formula for quantum dy- namical entropy of the Bogoliubov action of $\bigoplus Z_k, k=2,3,\ldots,$ on the $\text{CAR-}$algebra is obtained. It is proved that the part of action corresponding to the singular spectrum gives zero contribution to the entropy.