Surjective isometries between unitary sets of unital JB?-algebras
Surjective isometries between unitary sets of unital JB?-algebras
复制标题
酉 JB?-代数酉集之间的射射等距
DOI：
10.1016/j.laa.2022.02.003
复制
发表时间：
2022
影响因子：
1.1
通讯作者：
Cueto-Avellaneda M
中科院分区：
文献类型：
--
作者：
Cueto-Avellaneda M
在第一阶段,本文致力于建立一个topological-algebraic主成分的特征,U 0 (M),单一元素的集合,U (M),在unital JB⁎代数M .我们到达的结论,如unital C⁎代数,U 0 (M) = M 1−1∩U (M) = {U e h n⋯U e h 1 (1): n∈n、h j∈M s∀1≤≤n} = {U∈(M):存在w∈U 0 (M)为U−w为< 2}是分析弧连通。实际上，u0 (M)是U (M)的最小二次子集，包含集合e i M s a。我们的第二个目标是提供两个单位JB -代数M和n的主成分之间的满射等距的完整描述。与单位C -代数的情况相反，我们将推导出U (M)中作为度量空间不等距的连通成分的存在性。我们还将建立充射等距Δ： U (M)→U (N)允许扩展到M和N之间的充射线性等距的充分必要条件，这个结论并不总是成立的。在这些推论中，证明了M和N是Jordan -同构的当且仅当它们的主成分作为度量空间是等距的当，且仅当存在一个满射等距Δ： U (M)→U (N)将M的单位映射到U 0 (N)中的一个元素。这些结果为O. Hatori关于单位C -代数的结果的单位JB -代数的设置提供了推广。
This paper is, in a first stage, devoted to establishing a topological–algebraic characterization of the principal component, U 0 (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, U 0 (M)= M 1− 1∩ U (M)={U e i h n⋯ U e i h 1 (1): n∈ N, h j∈ M s a∀ 1≤ j≤ n}={u∈ U (M): there exists w∈ U 0 (M) with‖ u− w‖< 2} is analytically arcwise connected. Actually, U 0 (M) is the smallest quadratic subset of U (M) containing the set e i M s a. 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 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.