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

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 subset of M-1 and N subset of N-1 are clopen subsets of M-1 and N-1, respectively, which are closed for powers, inverses and products of the form U-a(b). In this paper we prove that for each surjective isometry Delta : M -> N there exists a surjective real-linear isometry T-0: M -> N and an element u(0) in the McCrimmon radical of N such that Delta(a) = T-0(a) + u(0) for all a is an element of M. Assuming that M and N are unital JB*-algebras we establish that for each surjective isometry Delta : M -> N the element Delta( 1) = u is a unitary element in Nand there exist a central projection p is an element of M and a complex-linear Jordan *-isomorphism Jfrom Monto the u*-homotope N-u* such that Delta(a) = J(p circle a) + J((1 - p) circle a*), for all a is an element of M. Under the additional hypothesis that there is a unitary element omega(0) in N satisfying U-omega 0(Delta(1)) = 1, we show the existence of a central projection p is an element of M and a complex-linear Jordan *-isomorphism Phi from M onto N such that Delta(a) = U-w0* (Phi(p circle a) + Phi((1 - p) circle a*)), for all a is an element of M. (C) 2021 The Author. Published by Elsevier Inc.