Isometries and Jordan-isomorphisms onto C*-algebras

Let A be a C*-algebra, and B a complex normed non-associative algebra. We p rove that, if B has an approximate unit bounded by one, then, for every lin ear isometry F from B onto A, there exists a Jordan-isomorphism G: B --> A and a unitary element u in the multiplier algebra of A such that F(x) = uG( x) for all x in B. We also prove that, if G is an isometric Jordan-isomorph ism from B onto A, then there exists a self-adjoint element phi in the cent re of the multiplier algebra of the closed ideal of A generated by the comm utators satisfying parallel to phi parallel to less than or equal to 1 and G(xy) = 1/2(G(x)G(y) + phi G(y)G(x) + (G(x)G(y) - G(y)G(x))) for all x, y in B.