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- W3166965836 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$^*$-algebra $M$. We arrive to the conclusion that, as in the case of unital C$^*$-algebras, $$begin{aligned}mathcal{U}^0(M) &= M^{-1}_{textbf{1}}capmathcal{U} (M) =leftlbrace U_{e^{i h_n}}cdots U_{e^{i h_1}}(textbf{1}) colon begin{array}{c} nin mathbb{N}, h_jin M_{sa} forall 1leq j leq n end{array} rightrbrace end{aligned}$$ 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$^*$-algebras $M$ and $N$. Contrary to the case of unital C$^*$-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 $^*$-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$^*$-algebras of the results obtained by O. Hatori for unital C$^*$-algebras." @default.
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- W3166965836 date "2021-05-31" @default.
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- W3166965836 title "Surjective isometries between unitary sets of unital JB$^*$-algebras" @default.
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