Isometries between JB*-triples

Let Z and W be JB*-triples and let T be a linear isometry from Z into W. For any z ∈ Z with ||z||<1, we show that Open image in new windowT(z) preserves the unit ball of T(Z). We show further that T is, locally, a triple homomorphism via a tripotent: for any z ∈ Z, there is a tripotent u in W** such that Open image in new windowa, b, c in the smallest subtriple Z _z of Z containing z, and also, {u,T(·),u}:Z _z →W** is an isometry.