Szymanski decompositions in von Neumann algebras

Abstract

A systematic study of the families of projections used in the proof of Szymanski's Decomposition Theorem for Operator-valued Functions is used to generalize that result to non-hereditary properties, to sets of operators or operator-valued functions, and to sets of properties. It is demonstrated that Szymanski's conditions in fact form a characterization of the most commonly studied case of parts-decomposition results. The new results are combined with generalizations of Kubo's definition of algebraically semi-definite properties of operators to produce a simple and unified proof of a large class of parts decompositions for operator-valued functions.