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

Abstract

Let Mand Nbe complex unital Jordan-Banach algebras, and let M−1and N−1denote the sets of invertible elements in Mand N, respectively. Suppose that M ⊆M−1and N ⊆N−1are clopen subsets of M−1and 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 →Nthere exists a surjective real-linear isometry T0:M→Nand an element u0in the McCrimmon radical of Nsuch that Δ(a) =T0(a) +u0for all a ∈M. Assuming that Mand Nare unital JB∗-algebras we establish that for each surjective isometry Δ :M →Nthe element Δ(1) =uis a unitary element in Nand there exist a central projection p ∈Mand a complex-linear Jordan ∗-isomorphism Jfrom Monto the u∗-homotope Nu∗such that Δ(a) = J(p ◦ a) + J((1 − p) ◦ a ∗), for all a ∈M. Under the additional hypothesis that there is a unitary element ω0in Nsatisfying Uω0(Δ(1)) =1, we show the existence of a central projection p ∈Mand a complex-linear Jordan ∗-isomorphism Φfrom Monto Nsuch that Δ(a) = Uw∗ 0 (Φ(p ◦ a) + Φ((1 − p) ◦ a ∗)) , for all a ∈M.