Jordan maps and pseudospectrum in C∗-algebras

We show that a surjective map φ between two unital C∗-algebras A and B, with φ(0)=0, satisfies Λε(φ(x1)−φ(x2))=Λε(x1−x2),(x1, x2∈A), where Λε denotes the ε-pseudospectrum, if and only if φ is a Jordan ∗-isomor\-phism. We also characterize maps φ1 and φ2 from A onto B that satisfy Λε(φ1(x1)φ2(x2))=Λε(x1x2),(x1, x2∈A), or some other binary operations, in terms of Jordan ∗-isomorphisms. The main results imply several other characterizations of Jordan ∗-isomorphisms which are interesting in their own right.