On a classical scheme in a noncommutative multiparameter ergodic theory

In the first part of the paper we describe the natural scheme for proving noncommutative individual ergodic theorems, generalize it for multiple sequences of measurable operators affiliated with a semifinite von Neumann algebra M, and apply it to theorems concerning unrestricted convergence of multiaverages. In the second part we prove convergence of ergodic averages induced by several maps satisfying specific recurrence relations, including so-called Multi Free Group Partial Sums. This is the multiindexed version of results obtained earlier jointly with V.I.Chilin and S.Litvinov.