Isometries between C*-algebras

Let A and B be C*-algebras and let T be a linear isometry from A into B. We show that there is a largest projection p in B∗∗ such that T (·)p : A −→ B∗∗ is a Jordan triple homomorphism and T (ab∗c + cb∗a)p = T (a)T (b)∗T (c)p + T (c)T (b)∗T (a)p for all a, b, c in A. When A is abelian, we have ‖T (a)p‖ = ‖a‖ for all a in A. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces locally to a Jordan triple isomorphism, by a projection.