2021/05/31 by Cueto-Avellaneda, María, Enami, Yuta, Hirota, Daisuke +2 · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2105.14870
This paper is, in a first stage, devoted to establish a topological--algebraic characterization of the principal component, U0 (M), of the set of unitary elements, U (M), in a unital JB^*-algebra M. We arrive to the conclusion that, as in the case of unital C^*-algebras, \beginalignedU0(M) amp;= M-11\capU (M) =\lbrace Uei hn⋯ Uei h1(1) \colon n∈ ℕ, hj∈ Msa ∀ 1≤ j ≤ n \rbrace \endaligned 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 U (M) which are not isometric as metric spaces. We shall also establish necessary and sufficient conditions to guarantee that a surjective isometry Δ: U(M)→ 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 Δ: U(M)→ U(N) mapping the unit of M to an element in U0(N). These results provide an extension to the setting of unital JB^*-algebras of the results obtained by O. Hatori for unital C^*-algebras.