Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries? Cueto Avellaneda, María Peralta Pereira, Antonio Miguel Isometry Jordan ∗-isomorphism Unitary set JB∗-algebra JBW*-algebra Extension of isometries First author supported by EPSRC (UK) project `Jordan Algebras, Finsler Geometry and Dynamics' ref. no. EP/R044228/1 and by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund project no. PGC2018-093332-B-I00, and Consejeria de Economia, Innovacion, Ciencia y Empleo, Junta de Andalucia grants FQM375 and A-FQM-242-UGR18. Second author supported by MCIN/AEI/10.13039/501100011033/FEDER `Una manera de hacer Europa' project no. PGC2018-093332-B-I00, Consejeria de Economia, Innovacion, Ciencia y Empleo, Junta de Andalucia grants FQM375, A-FQM-242-UGR18 and PY20_00255, and by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. 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. 2021-12-16T13:24:38Z 2021-12-16T13:24:38Z 2020-05-10 info:eu-repo/semantics/article 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] http://hdl.handle.net/10481/72097 10.1080/03081087.2021.2003745 eng http://creativecommons.org/licenses/by-nc-nd/3.0/es/ info:eu-repo/semantics/openAccess Atribución-NoComercial-SinDerivadas 3.0 España Taylor & Francis