2022/05/23 by Cabezas, David, Peralta, Antonio M.
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2205.11176
It is known, by Gelfand theory, that every commutative JB^*-triple admits a representation as a space of continuous functions of the form C0^\mathbbT(L) = \ a∈ C0(L) : a(λt ) = λa(t), ∀ λ∈ \mathbbT, t∈ L\, where L is a principal \mathbbT-bundle and \mathbbT denotes the unit circle in ℂ. We provide a description of all orthogonality preserving (non-necessarily continuous) linear maps between commutative JB^*-triples. We show that each linear orthogonality preserver T: C0^\mathbbT (L1)→ C0^\mathbbT (L2) decomposes in three main parts on its image, on the first part as a positive-weighted composition operator, on the second part the points in L2 where the image of T vanishes, and a third part formed by those points s in L2 such that the evaluation mapping δs∘ T is non-continuous. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB^*-triples is automatically continuous and biorthogonality preserving.