LINEAR ISOMETRIES BETWEEN REAL JB*-TRIPLES AND C*-ALGEBRAS

Abstract

Let T : A ! B be a (not necessarily surjective) linear isometry between two real JB ∗ -triples. Then for each a 2 A there exists a tripotent ua in the bidual, B '' , of B such that (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; (b) The mapping {ua,T(·),ua} : Aa ! B '' is a linear isometry. Furthermore, when B is a real C ∗ -algebra, the projection p = pa = u ∗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 2 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.