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

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