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 $\mathfrak{M}\subseteq M^{-1}$ 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 $\mathrm{\Delta }:\mathfrak{M}\to \mathfrak{N}$ 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 $\mathrm{\Delta }(a)=T_0(a)+u_0$ for all $a\in \mathfrak{M}$. Assuming that M and N are unital ${\mathrm{JB}}^{⁎}$-algebras we establish that for each surjective isometry $\mathrm{\Delta }:\mathfrak{M}\to \mathfrak{N}$ the element $\mathrm{\Delta }(\mathbf{\text{1}})=u$ 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

$$\mathrm{\Delta }(a)=J(p\circ a)+J((\mathbf{\text{1}}-p)\circ a^{⁎}),$$

for all $a\in \mathfrak{M}$. Under the additional hypothesis that there is a unitary element ${\omega }_0$ in N satisfying $U_{{\omega }_0}(\mathrm{\Delta }(\mathbf{\text{1}}))=\mathbf{\text{1}}$, we show the existence of a central projection $p\in M$ and a complex-linear Jordan ^⁎-isomorphism Φ from M onto N such that

$$\mathrm{\Delta }(a)=U_{w_0^{⁎}}(\mathrm{\Phi }(p\circ a)+\mathrm{\Phi }((\mathbf{\text{1}}-p)\circ a^{⁎})),$$

for all $a\in \mathfrak{M}$.