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

Abstract

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).

Commutativity of projections and characterization of traces on Von Neumann algebras

“…In [3] we gave another proof of this fact with a uniform estimate for the number of summands in this representation. The least upper bound of three factors is related to the existence of a nontrivial finite trace on these algebras.…”

“…Denote by B(H ) the * -algebra of all bounded linear operators in H . Given x ∈ B(H ), denote by σ(x) the spectrum of x, and put |x| = (x * x) 1 …”

“…Suppose that x ≤ 1 and put p = diag(1, 0). Define two projections r (1,a) and r (1,1−b) by the operator 2 × 2 matrices using (2). Then uxu * = pr (1,a) …”

“…In [2] we proved an unimprovable statement (with respect to the number of factors): If a von Neumann algebra M has no abelian direct summands (is properly infinite) then we can express every operator x ∈ M as a finite sum x =…”

Sums of compositions of pairs of projections

“…Bikchentaev generalized his result about representation x = n i=1 Q i P i in B(H) to wide classes of C * -algabras, in particular he considered properly infinite von Neumann algebras ( [2], [3]). We believe that all the results proved in our paper can also be generalized from B(H) to any properly infinite von Neumann algebra.…”