Piecewise ⁎-homomorphisms and Jordan maps on C⁎-algebras and factor von Neumann algebras

We investigate maps between C⁎-algebras that are well behaved with respect to mutually commuting elements. We contribute to the Mackey-Gleason problem by showing that any continuous bijection between self-adjoint parts of C⁎-algebras that preserves triple product (a,b)→aba, and is linear on commutative subspaces, is already linear. This allows us to describe such maps as direct differences of linear Jordan isomorphisms. We shall show that any weak⁎-continuous bijection between positive invertible elements of von Neumann factors (of dimension at least 9) that preserves products of commuting elements in both directions is of the form a→eψ(log⁡a)θ(ac), where θ is a linear Jordan ⁎-isomorphism, c nonzero real number and ψ is a hermitian continuous functional. In a similar way we describe the same type of bicontinuous maps between unitary groups of von Neumann factors. General form of the above mentioned maps on C⁎-algebras is also presented.