Commutativity Preserving Mappings of Von Neumann Algebras

Abstract

AbstractA map θ: M —> N where M and N are rings is said to preserve commutativity in both directions if the elements a,b ∊ M commute if and only if θ(a) and θ(b) commute. In this paper we show that if M and N are von Neumann algebras with no central summands of type I1 or I2 and θ is a bijective additive map which preserves commutativity in both directions then θ(x) = cφ(x) +f(x) where c is an invertible element in ZN, the center of N, φ M —> N is a Jordan isomorphism of M onto N, and f is an additive map of M into ZN.