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The paper deals with spectral properties of abelian dynamical semigroups on von Neumann algebras. The notions of an eigenvalue and eigenspace for such semigroups are defined, generalizing those known for the classical case, and the existence of normal projections on the eigenspaces is proved. A more detailed study of properties of these projections is performed. Also spectral properties of tensor products of completely positive dynamical semigroups are investigated.

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The paper is devoted to the investigation of the notion of sufficiency in quantum statistics. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), Petz's sufficiency, and Umegaki's sufficiency. The problem of the existence and structure of the minimal sufficient subalgebra is analyzed in some detail, conditions yielding equivalence of the three modes of sufficiency are considered, and quantum Basu's theorem is obtained. Moreover, it is shown that an interesting "factorization theorem" of Jenčová and Petz needs some corrections to hold true.