Surjective isometries between sets of invertible elements in unital Jordan-Banach algebras

Let M and N be complex unital Jordan-Banach algebras, let M−1 N−1 denote the sets of invertible elements in N, respectively. Suppose that M⊆M−1 N⊆N−1 are clopen subsets N−1, respectively, which closed for powers, inverses products form Ua(b). In this paper we prove each surjective isometry Δ:M→N there exists a real-linear T0:M→N an element u0 McCrimmon radical such Δ(a)=T0(a)+u0 all a∈M. Assuming JB⁎-algebras establish Δ(1)=u is unitary exist central projection p∈M complex-linear Jordan ⁎-isomorphism J from onto u⁎-homotope Nu⁎ thatΔ(a)=J(p∘a)+J((1−p)∘a⁎), Under additional hypothesis ω0 satisfying Uω0(Δ(1))=1, show existence Φ thatΔ(a)=Uw0⁎(Φ(p∘a)+Φ((1−p)∘a⁎)),