Linear isometries between real JB*-triples and C*-algebras

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Let T: A→ B be a (not necessarily surjective) linear isometry between two real JB*-triples. Then for each a∈ A there exists a tripotent uₐ in the bidual, B'', of B such that enumerate[(a)] {uₐ,T({f,g,h}),uₐ}={uₐ,{T(f),T(g),T(h)},uₐ}, for all f,g,h in the real JB*-subtriple, Aₐ, generated by a; The mapping {uₐ,T(·),uₐ}:Aₐ→ B'' is a linear isometry. enumerate Furthermore, when B is a real C*-algebra, the projection p=pₐ= uₐ* uₐ satisfies that T(·)p:Aₐ→ B'' is an isometric triple homomorphism. When A and B are real C*-algebras and A is abelian of real type, then there exists a partial isometry u∈ B'' such that the mapping T(·)u*u:A→ B'' is an isometric triple homomorphism. These results generalise, to the real setting, some previous contributions due to C.-H. Chu and N.-C. Wong, and C.-H. Chu and M. Mackey in 2004 and 2005. We give an example of a non-surjective real linear isometry which cannot be complexified to a complex isometry, showing that the results in the real setting can not be derived by a mere complexification argument.