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International Journal of Mathematics and Mathematical Sciences
Volume 9, Issue 4, Pages 767-770
http://dx.doi.org/10.1155/S0161171286000923

On the operator equation α+α1=β+β1

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

Received 25 May 1985; Revised 1 July 1986

Copyright © 1986 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let α,β be -automorphisms of a von Neumann algebra M satisfying the operator equation α+α1=β+β1. In this paper we use new techniques (which are useful in noncommutative situations as well) to provide alternate proofs of the results:- If α,β commute then there is a central projection p in M such that α=β on MP and α=β1 on M(1P); If M=B(H), the algebra of all bounded operators on a Hilbert space H, then α=β or α=β1.