On a class of operators in von Neumann algebras with Segal measure on the projectors

By means of the concept of Segal measure, defined on projectors (and thus on subspaces associated with a von Neumann algebra) we introduce the concept of relative compactness of sets and, on this basis, the concept of operators completely continuous with respect to the von Neumann algebra and the Segal measure. The article is concerned with the formal structure of the theory of this class of operators: the general theorem of Calkin is obtained on the uniqueness of the ideal with respect to completely continuous operators; a theory is constructed for perturbations of Hermitian operators with respect to completely continuous ones; singular and characteristic numbers are introduced for operators from the von Neumann algebra and their minimax properties are derived; some characterizations are introduced in terms of completely continuous operators.
Bibliography: 10 titles.