The results on noncommutative ergodic theory are proved in the following setting: M is a -algebra, a semigroup of strongly positive (i.e. linear maps of M into itself (no continuity assumptions of as a function of t is required), and is a faithful - invariant normal state on M. It is shown that many results known in the case when is a group of *-automorphisms [O. Bratteli and the author, Operator algebras and quantum statistical mechanics, Vol. I (1979; Zbl 0421.46048)] can be extended to this situation. Some results of the paper [A. Frigerio, ibid. 63, 269-276 (1978; Zbl 0404.46050)] are also generalized.
Among the results obtained in the paper are:
i) a description of the set of invariant elements in M;
ii) conditions that an invariant state have a unique decomposition into ergodic states;
iii) a criterium of ergodicity of ;
iv) in the case when is 2-positive, a strong positivity of a semigroup is proved, where is given by ( is the cyclic and separating vector associated with and sends into , . It is shown that -ergodicity of is equivalent to uniform clustering property with respect to : for all normal states ’.