Linear orthogonality preservers between function spaces associated with commutative JB${}^{\ast }$-triples

Abstract

It is known, by Gelfand theory, that every commutative JB${}^{\ast }$-triple admits a representation as a space of continuous functions of the form $$C_0^{\mathbb{T}}(L)=\{a\in C_0(L):a(\lambda t)=\lambda a(t),\forall \lambda \in \mathbb{T},t\in L\},$$ where $L$ is a principal $\mathbb{T}$-bundle and $\mathbb{T}$ denotes the unit circle in $\mathbb{C}.$ We provide a description of all orthogonality preserving (non-necessarily continuous) linear maps between commutative JB${}^{\ast }$-triples. We show that each linear orthogonality preserver $T:C_0^{\mathbb{T}}(L_1)\to C_0^{\mathbb{T}}(L_2)$ 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 $L_2$ where the image of $T$ vanishes, and a third part formed by those points $s$ in $L_2$ such that the evaluation mapping ${\delta }_s\circ T$ is non-continuous. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB${}^{\ast }$-triples is automatically continuous and biorthogonality preserving.