ABSTRACT. A characterization of the traces in a broad class of weights on yon 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 yon Neumann algebras and to the K5the duality theory for ideal spares of Segal measurable operators are given.KEY woRDS: von Neumann algebra, affiliated operator, trace, weight, Jordan algebra, logic of orthoprojections, measure on projections, noncommutative integration, KSthe duality.w Characterization of traces on yon Neumann algebras Let M be a yon Neumann algebra acting on a Hilbert space H. We denote by M erm , M uni , M + , and M'P r~ its Hermitian, unitary, and positive part, and the logic of orthoprojections, respectively. Next, let e be the identity element of M and II 9 II the C'-no~m on M.

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.Let H be a Hilbert space over the field C. Denote by B(H ) the * -algebra of all bounded linear operators in H . An operatorIn this article we solve the following problems: (I) Represent each element of a von Neumann algebra M having no direct abelian summand as a finite sum of finite products of projections in M .(II) Find the least upper bound for the number of product terms in the summands of such representations.Earlier, other authors [1-7] only considered the representations in a weaker form: the summands were allowed to have coefficients in C (or R), while problem (II) was not studied at all. In [8] the author obtained a complete solution to (I) and (II) in the case of M = B(H ): each bounded linear operator x in a complex Hilbert space H is representable as a finite sum x = x k in which each x k is the product of at most two projections for dim H = ∞ and at most three projections for 2 ≤ dim H < ∞.In § 3 we present necessary facts and give a brief survey of the results on this topic. In § 4 we prove an auxiliary assertion which is also of interest in its own right. In § 5 we prove the following unimprovable assertion (as regards the number of product terms): if a von Neumann algebra M has no direct abelian summand (is properly infinite) then each operator x ∈ M is representable as a finite sum x = x k in which each x k is a product of at most three (two) projections in M .For a von Neumann algebra without a direct abelian summand, the proof bases on a new representation for operators in the form of finite sums of pairwise products of projections and idempotents. It is interesting that the least upper bound in question is connected with existence of a nontrivial finite trace on these algebras. As a consequence, our result gives a new proof of equivalence of the primary classification of von Neumann algebras in terms of projections [9] and in terms of traces [10] and also a description for the Jordan structure of the Hermitian part of a von Neumann algebra (the "algebra of observables" of quantum mechanics) in terms of projections (i.e., the "questions" of quantum mechanics).

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