Linear orthogonality preservers of Hilbert bundles

Abstract

Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $θ$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if $\left<θ(x),θ(y)\right> =0$ whenever $\left =0$. We prove that if $θ$ is an orthogonality preserving map from a full Hilbert $C_0(Ω)$-module $E$ into another Hilbert $C_0(Ω)$-module $F$ that satisfies a weaker notion of $C_0(Ω)$-linearity (known as "localness"), then $θ$ is bounded and there exists $ϕ\in C_b(Ω)_+$ such that $$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left , \quad \forall x,y \in E. $$ On the other hand, if $F$ is a full Hilbert $C^*$-module over another commutative $C^*$-algebra $C_0(Δ)$, we show that a "bi-orthogonality preserving" bijective map $θ$ with some "local-type property" will be bounded and satisfy $$ \left<θ(x),θ(y)\right>\ =\ ϕ\cdot\left \circσ, \quad \forall x,y \in E $$ where $ϕ\in C_b(Ω)_+$ and $σ: Δ\rightarrow Ω$ is a homeomorphism.