Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?
Metadatos
Mostrar el registro completo del ítemEditorial
Taylor & Francis
Materia
Isometry Jordan ∗-isomorphism Unitary set JB∗-algebra JBW*-algebra Extension of isometries
Fecha
2020-05-10Referencia bibliográfica
Published version: María Cueto-Avellaneda & Antonio M. Peralta (2021) Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?, Linear and Multilinear Algebra, DOI: [10.1080/03081087.2021.2003745]
Patrocinador
EPSRC (UK) project `Jordan Algebras, Finsler Geometry and Dynamics' EP/R044228/1; Spanish Ministry of Science, Innovation and Universities (MICINN); European Commission PGC2018-093332-B-I00; Junta de Andalucia FQM375 A-FQM-242-UGR18 PY20_00255; IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033 MCIN/AEI/10.13039/501100011033/FEDERResumen
Let M and N be two unital JB*-algebras and let U(M) and U(N) denote the sets of all unitaries in M and N, respectively. We prove that the following statements are equivalent:
(a) M and N are isometrically isomorphic as (complex) Banach spaces;
(b) M and N are isometrically isomorphic as real Banach spaces;
(c) there exists a surjective isometry Delta : U(M) -> U(N).
We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry Delta : U(M) -> U(N), we can find a surjective real linear isometry Psi : M -> N which coincides with Delta on the subset e(iMsa). If we assume that M and N are JBW*-algebras, then every surjective isometry Delta : U(M) -> U(N) admits a (unique) extension to a surjective real linear isometry from M onto N. This is an extension of the Hatori-Molnar theorem to the setting of JB*-algebras.