A quantitative version of Herstein's theorem for Jordan *-isomorphisms

We study linear mappings between C*-algebras A and B, which approximately satisfy Jordan multiplicativity condition and a *-preserving condition (that is, the so-called e-approximate Jordan *-homomorphisms). We first prove that every such mapping is automatically continuous and we give the estimates of its norm, as well as the estimates of the norm of its inverse if it is bijective. IfK(H-1) subset of A subset of B(H-1), K(H-2) subset of B subset of B(H-2), and psi : A -> B is a bijective epsilon-approximate Jordan *-homomorphism with sufficiently small epsilon > 0, then either psi(-1) has a large norm, or psi is close to a Jordan *-isomorphism, that is, to a mapping of the form X -> UXU*, or X -> (UXU)-U-t*, for some unitary U is an element of B(H-1, H-2). We also give the corresponding quantitative estimate.