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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.

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Authors and Affiliations

  1. Chebotarev Research Institute of Mathematics and Mechanics, Kazan’ State University, Russia

    A. M. Bikchentaev

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  1. A. M. Bikchentaev
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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.

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Bikchentaev, A.M. On representation of elements of a von Neumann algebra in the form of finite sums of products of projections. Sib Math J 46, 24–34 (2005). https://doi.org/10.1007/s11202-005-0003-4

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  • DOI: https://doi.org/10.1007/s11202-005-0003-4

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