ISOMETRIES AND JORDAN-ISOMORPHISMS ONTO C∗-ALGEBRAS

Abstract. Let A be a C∗-algebra, and B a complex normed non-associative algebra. We prove that, if B has an approximate unit bounded by one, then, for every linear 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-isomorphism from B onto A, then there exists a self-adjoint element ϕ in the centre of the multiplier algebra of the closed ideal of A generated by the commutators satisfying ‖ϕ ‖ 6 1 and G(xy) = 1 2 (G(x)G(y) +G(y)G(x) + ϕ(G(x)G(y)−G(y)G(x))