Additive mappings on von Neumann algebras preserving absolute values

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