The structure of linear zero product and commutativity preservers of C∗-algebras.

Let θ : A sa → B sa be a bijective real linear zero product preserver between the self-adjoint parts of two (complex) C ∗ -algebras. Suppose θ preserves commutativity in two directions (i.e., a b = b a if and only if θ (a) θ (b) = θ (b) θ (a) ). We show that θ is automatically continuous. More precisely, there is an invertible central self-adjoint multiplier h of B , and a real linear Jordan isomorphism J : A sa → B sa such that θ = h J . [ABSTRACT FROM AUTHOR]