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

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.