Surjective isometries between unitary sets of unital JB${}^{\ast }$-algebras

Abstract

This paper is, in a first stage, devoted to establish a topological--algebraic characterization of the principal component, ${\mathcal{U}}^0(M)$, of the set of unitary elements, $\mathcal{U}(M)$, in a unital JB${}^{\ast }$-algebra $M$. We arrive to the conclusion that, as in the case of unital C${}^{\ast }$-algebras, $$\begin{array}{rl}{\mathcal{U}}^0(M)& =M_{\text{1}}^{-1}\cap \mathcal{U}(M)=\left\{U_{e^{i{h}_n}}\cdots U_{e^{i{h}_1}}(\text{1}):\begin{array}{c}n\in \mathbb{N},h_j\in M_{sa}\forall 1\le j\le n\end{array}\right\}\end{array}$$ is analytically arcwise connected. Our second goal is to provide a complete description of the surjective isometries between the principal components of two unital JB${}^{\ast }$-algebras $M$ and $N$. Contrary to the case of unital C${}^{\ast }$-algebras, we shall deduce the existence of connected components in $\mathcal{U}(M)$ which are not isometric as metric spaces. We shall also establish necessary and sufficient conditions to guarantee that a surjective isometry $\Delta :\mathcal{U}(M)\to \mathcal{U}(N)$ admits an extension to a surjective linear isometry between $M$ and $N$, a conclusion which is not always true. Among the consequences it is proved that $M$ and $N$ are Jordan ${}^{\ast }$-isomorphic if, and only if, their principal components are isometric as metric spaces if, and only if, there exists a surjective isometry $\Delta :\mathcal{U}(M)\to \mathcal{U}(N)$ mapping the unit of $M$ to an element in ${\mathcal{U}}^0(N)$. These results provide an extension to the setting of unital JB${}^{\ast }$-algebras of the results obtained by O. Hatori for unital C${}^{\ast }$-algebras.