Linear orthogonality preservers between function spaces associated with commutative JB$^*$-triples

TL;DR

This paper characterizes all orthogonality-preserving linear maps between commutative JB*-triples, showing they decompose into weighted composition operators and analyzing their continuity and bijective properties.

Contribution

It provides a complete description of orthogonality-preserving linear maps between commutative JB*-triples, including their decomposition and continuity properties.

Findings

Orthogonality preservers decompose into weighted composition operators.

Every bijective orthogonality preserver is automatically continuous.

The structure involves parts where the map vanishes or is non-continuous.

Abstract

It is known, by Gelfand theory, that every commutative JB$^*$-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$^*$-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$^*$-triples is automatically continuous and biorthogonality preserving.