The structure of linear zero product and commutativity preservers of $$C^*$$-algebras

Let $$\theta :\mathcal {A}_{sa}\rightarrow \mathcal {B}_{sa}$$ be a bijective real linear zero product preserver between the self-adjoint parts of two (complex) $$C^*$$ -algebras. Suppose $$\theta $$ preserves commutativity in two directions (i.e., $$ab=ba$$ if and only if $$\theta (a)\theta (b)=\theta (b)\theta (a)$$ ). We show that $$\theta $$ is automatically continuous. More precisely, there is an invertible central self-adjoint multiplier h of $$\mathcal {B}$$ , and a real linear Jordan isomorphism $$J: \mathcal {A}_{sa}\rightarrow \mathcal {B}_{sa}$$ such that $$\theta = hJ$$ .