Surjective isometries between unitary sets of unital JB∗-algebras
Metadatos
Mostrar el registro completo del ítemAutor
Cueto Avellaneda, María; Enami, Yuta; Hirota, Daisuke; Miura, Takeshi; Peralta Pereira, Antonio MiguelEditorial
Elsevier
Materia
Isometry Unitary Extension of isometries
Fecha
2022-06-15Referencia bibliográfica
M. Cueto-Avel laneda et al. Surjective isometries between unitary sets of unital JB∗-algebras. . [https://doi.org/10.1016/j.laa.2022.02.003]
Patrocinador
CBUA; Consejería de Economía y Conocimiento de la Junta de Andalucía A-FQM-242-UGR18, FQM375; Ministerio de Ciencia, Innovación y Universidades; Engineering and Physical Sciences Research Council EP/R044228/1; Universidad de Granada; Ministerio de Ciencia e Innovación; Japan Society for the Promotion of Science JP 20K03650, JP 21J21512; European Regional Development Fund PGC2018-093332-B-I00Resumen
This paper is, in a first stage, devoted to establishing 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, U0(M)=M1−1∩U(M)={Ue⋯Ue(1):n∈N,hj∈Msa∀1≤j≤n}={u∈U(M): there exists w∈U0(M) with ‖u−w‖<2} is analytically arcwise connected. Actually, U0(M) is the smallest quadratic subset of U(M) containing the set eiM. 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.