Dynamical entropy for Bogoliubov actions of torsion-free abelian groups on the CAR-algebra Valentin Ya. Golodets Sergey V. Neshveyev Abstract We compute dynamical entropy in Connes, Narnhofer and Thirring sense for a Bogoliubov action of a torsion-free abelian group G on the CAR-algebra. A formula analogous to that found by Størmer and Voiculescu in the case G = Z is obtained. The singular part of a unitary representation of G is shown to give zero contribution to the entropy. A proof of these results requires new arguments since a torsion-free group may have no finite index proper subgroups. Our approach allows to overcome these difficulties, it differs from that of Størmer-Voiculescu. Introduction. Entropy is an important notion in classical statistical mechanics and information theory. Initially the notion of entropy for automorphisms of a measure space was introduced by Kolmogorov and Sinai in 1958. This invariant proved to be extremely useful, it generated an entire field in the theory of classical dynamical systems and topological dynamics. The extension of the notion of entropy onto quantum systems was treated as a difficult math- ematical problem. It was solved by Connes and Størmer [CS] only in 1975 for dynamical systems of type II 1 . Then Connes, Narnhofer and Thirring [CNT] extended this theory to general C ∗ - and W ∗ -dynamical systems. The computation of dynamical entropy for specific models is one of the principal trends in the theory (see [BG],[GN] for a bibliography). One of the main results in this sphere belongs to Størmer and Voiculescu [StV]. They showed that the CNT-entropy of a Bogoli- ubov automorphism of the CAR-algebra is computed by a simple formula (predicted by A. Connes for the tracial state), and only the absolutely continuous part of the unitary oper- ator defining the Bogoliubov automorphism gives a contribution to the entropy. Bezuglyi and Golodets [BG] obtained the same results for Bogoliubov actions of free abelian groups. It is quite natural to extend these results to Bogoliubov actions of arbitrary countable torsion-free abelian groups. Note that in Størmer-Voiculescu’s approach it is very important that the group Z has a lot of finite index subgroups (see Theorem 2.1, condition (iv), in [StV]). But, for example, the group Q of rational numbers contains no finite index (proper) subgroups. So the methods of [StV] and [BG] cannot be immediately applied. It is interesting to note that the problem of studying entropic properties of actions of the group Q is well-known in the commutative entropic theory. As far as we know there are no methods to describe the Pinsker algebra and asymptotic properties of systems with completely positive entropy for such actions. In particular, the Conze approach [Co] does not allow to solve these problems. It strengthens our interest to Bogoliubov actions of torsion-free abelian groups. 1