On the operator equation α + α − 1 = β + β − 1

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 M P 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 .