On a property of $L_p$ spaces on semifinite von Neumann algebras

A characterization of the traces in a broad class of weights on von Neumann algebras is obtained. A new property of the “domain ideals” of these traces is proved. In the semifinite case, a relation for a faithful normal trace is established. This result is new even for the algebra of all bounded operators on a Hilbert space. Applications of the main result to the structure theory of von Neumann algebras and to the Köthe duality theory for ideal spaces of Segal measurable operators are given.