On pairs of automorphisms of von Neumann algebras (Q1826079)

Let a and b be *-automorphisms of a von Neumann algebra M satisfying the operator equation \(a+a^{-1}=b+b^{-1}.\) It was known that if a and b commute then there exists a central projection p in M such that \(a=b\) on Mp and \(a=b^{-1}\) on M(I-p). In this paper, an example is given to show that the above statement may not be true when a and b do not commute. However, it is proved that if a and b satisfy the equation, then there exists a central projection p in M such that \(a^ 2=b^ 2\) on Mp and \(a^ 2=b^{-2}\) on M(I-p). This provides a general solution of the equation for von Neumann algebras.