On representation of elements of a Von Neumann algebra in the form of finite sums of products of projections

We prove that each element of the von Neumann algebra without a direct abelian summand is representable as a finite sum of products of at most three projections in the algebra. In a properly infinite algebra the number of product terms is at most two. Our result gives a new proof of equivalence of the primary classification of von Neumann algebras in terms of projections and traces and also a description for the Jordan structure of the “algebra of observables” of quantum mechanics in terms of the “questions” of quantum mechanics.