International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 4, Pages 767-770
doi:10.1155/S0161171286000923
On the operator equation α+α−1=β+β−1
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
Received 25 May 1985; Revised 1 July 1986
Copyright © 1986 A. B. Thaheem. 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+α−1=β+β−1. 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=β−1 on M(1−P); If M=B(H), the algebra of all bounded operators on a Hilbert space H, then α=β or α=β−1.