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

Abstract

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