On representation of elements of a von Neumann algebra in the form of finite sums of products of projections
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- Bikchentaev, A.M. Sib Math J (2005) 46: 24. doi:10.1007/s11202-005-0003-4
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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.