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

2013/09/16 by Maria Apazoglou, Apazoglou, Maria, Antonio M. Peralta +1
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #math.OA

paper · pdf · doi:10.48550/arxiv.1309.3838

to appear in Quart. J. Math

arxiv created 2013/09/16 · arxiv updated 2013/09/17

Abstract

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 u_a in the bidual, B'', of B such that \beginenumerate[(a)] \item \u_a,T(\f,g,h\),u_a\=\u_a,\T(f),T(g),T(h)\,u_a\, for all f,g,h in the real JB^*-subtriple, A_a, generated by a; \item The mapping \u_a,T(⋅),u_a\ :A_a→ B'' is a linear isometry. \endenumerate Furthermore, when B is a real C^*-algebra, the projection p=p_a= u_a^* u_a satisfies that T(⋅)p :A_a→ 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.

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