Siberian Mathematical Journal

, 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

  • A. M. Bikchentaev
Article

DOI: 10.1007/s11202-005-0003-4

Cite this article as:
Bikchentaev, A.M. Sib Math J (2005) 46: 24. doi:10.1007/s11202-005-0003-4

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.

Keywords

C*-algebravon Neumann algebratracebounded linear operatoridempotentprojectionlinear spanHilbert space

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. M. Bikchentaev
    • 1
  1. 1.Chebotarev Research Institute of Mathematics and MechanicsKazan’ State UniversityRussia