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${}^{\ast }$-algebras we establish that for each surjective isometry $\Delta :\mathfrak{M}\to \mathfrak{N}$ the element $\Delta (\text{1})=u$ is a unitary element in $N$ and there exist a central projection $p\in M$ and a complex-linear Jordan ${}^{\ast }$-isomorphism $J$ from $M$ onto the $u^{\ast }$-homotope $N_{u^{\ast }}$ such that $$\Delta (a)=J(p\circ a)+J((\text{1}-p)\circ a^{\ast }),$$ for all $a\in \mathfrak{M}$. Under the additional hypothesis that there is a unitary element ${\omega }_0$ in $N$ satisfying $U_{{\omega }_0}(\Delta (\text{1}))=\text{1}$, we show the existence of a central projection $p\in M$ and a complex-linear Jordan ${}^{\ast }$-isomorphism $\Phi$ from $M$ onto $N$ such that $$\Delta (a)=U_{w_0^{\ast }}\left(\Phi (p\circ a)+\Phi ((\text{1}-p)\circ a^{\ast })\right),$$ for all $a\in \mathfrak{M}$.