2020/05/10 by María Cueto-Avellaneda, Cueto-Avellaneda, María, Antonio M. Peralta +1 · 1 citation
Mathematics · #17C65 #46A22 #46B03 #46B20 #46H70 Secondary 46B04 #46L05 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 47B49
paper · pdf · doi:10.48550/arxiv.2005.04794
openalex publication_date 2020/05/10 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
Let M and N be two unital JB^*-algebras and let \U (M) and\n\U (N) denote the sets of all unitaries in M and N,\nrespectively. We prove that the following statements are equivalent:\n (a) M and N are isometrically isomorphic as (complex) Banach spaces;\n (b) M and N are isometrically isomorphic as real Banach spaces;\n (c) There exists a surjective isometry \Δ: \U(M)\→\n\U(N).\n We actually establish a more general statement asserting that, under some\nmild extra conditions, for each surjective isometry \Δ:\U (M) \→\n\U (N) we can find a surjective real linear isometry \Ψ:M\→ N\nwhich coincides with \Δ on the subset e^i Msa. If we assume that\nM and N are JBW^*-algebras, then every surjective isometry\n\Δ:\U (M) \→ \U (N) admits a (unique) extension to a\nsurjective real linear isometry from M onto N. This is an extension of the\nHatori--Moln 'ar theorem to the setting of JB^*-algebras.\n