On classification of von Neumann algebras in a space with conjugation

Abstract

In this paper for the first time we show that in the complex Hilbert space with the conjugation operator a classification of von Neumann algebras is possible. Similar classification is known for Krein spaces. Projectors (idempotents) often serve as elements of quantum logic. In operator theories projectors play the role of elements from which bounded operators are constructed. For one special case we show that for any projector from von Neumann algebra which acts in a separable Hilbert space one can always find conjugation operator J adjoined to this algebra for which the projector is self-adjoint.