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

Abstract

Let $M$ and $N$ be unital Jordan-Banach algebras, and let $M^{-1}$ and $N^{-1}$ denote the sets of invertible elements in $M$ and $N$, respectively. Suppose that $\mathfrak{M}\subseteq M^{-1}$ and $\mathfrak{N}\subseteq N^{-1}$ 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 $\Delta : \mathfrak{M}\to \mathfrak{N}$ there exists a surjective real-linear isometry $T_0: M\to N$ and an element $u_0$ in the McCrimmon radical of $N$ such that $\Delta (a) = T_0(a) +u_0$ for all $a\in \mathfrak{M}$.\smallskip Assuming that $M$ and $N$ are unital JB$^*$-algebras we establish that for each surjective isometry $\Delta : \mathfrak{M}\to \mathfrak{N}$ the element $\Delta(\textbf{1}) =u$ is a unitary element in $N$ and there exist a central projection $p\in M$ and a complex-linear Jordan $^*$-isomorphism $J$ from $M$ onto the $u^*$-homotope $N_{u^*}$ such that $$\Delta (a) = J(p\circ a) + J ((\textbf{1}-p) \circ a^*),$$ for all $a\in \mathfrak{M}$. Under the additional hypothesis that there is a unitary element $\omega_0$ in $N$ satisfying $U_{\omega_0} (\Delta(\textbf{1})) = \textbf{1}$, we show the existence of a central projection $p\in M$ and a complex-linear Jordan $^*$-isomorphism $\Phi$ from $M$ onto $N$ such that $$\Delta (a) = U_{w_0^{*}} \left(\Phi (p\circ a) + \Phi ((\textbf{1}-p) \circ a^*)\right),$$ for all $a\in \mathfrak{M}$.