Additive mappings on von Neumann algebras preserving absolute values

AbstractGiven Hilbert spaces H and K and a von Neumann algebra A⊂B(H), let Φ denote the class of all additive mappings ϕ:A→B(K) satisfying |ϕ(A)|=ϕ(|A|) (A∈A). The paper shows that if A contains no nonzero abelian central projection then every ϕ∈Φ preserves the *-operation, the R-linear combination, and, up to a commuting operator multiple ϕ(I)⩾0, the (ring) multiplication. If A contains a nonzero abelian central projection P and if the dimension of K is at least 2 or 2rank(P) according to whether or not P can be chosen to be minimal, then there exists an additive mapping ϕ:A→B(K) such that ϕ(I) is a projection and |ϕ(A)|=ϕ(|A|) for all A∈A but ϕ is neither multiplicative nor adjoint preserving. In case A=B(H) the result was proved by Molnár [Bull. Aust. Math. Soc. 53 (1996) 391] when ϕ(A) contained all finite rank operators, and by Radjabalipour et al. [Linear Algebra Appl. 327 (2001) 197] under the (redundant) restriction ϕ(iI)K⊂ϕ(I)K