, Volume 46, Issue 1, pp 24-34

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

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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.

Original Russian Text Copyright © 2005 Bikchentaev A. M.
The author was supported by the Program “Universities of Russia” (Grant UR.04.01.011).
Translated from Sibirski \( \overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{\imath} \) Matematicheski \( \overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{\imath} \) Zhurnal, Vol. 46, No. 1, pp. 32–45, January–February, 2005.