Linear Orthogonality Preservers of Hilbert Modules

We verify in this paper that the linearity and orthogonality structures of a (not necessarily local trivial) Hilbert bundle over a locally compact Hausdorff space Ω determine its unitary structure. In fact, as Hilbert bundles over Ω are exactly Hilbert C0(Ω)-modules, we have a more general set up. A C-linear map θ (not assumed to be bounded) between two Hilbert C∗-modules is said to be “orthogonality preserving” if 〈θ(x), θ(y)〉 = 0 whenever 〈x, y〉 = 0. We prove that if θ is a orthogonality preserving C0(Ω)-module map from a full Hilbert C0(Ω)module E into another Hilbert C0(Ω)-module F , then θ is bounded and there exists φ ∈ Cb(Ω)+ such that 〈θ(x), θ(y)〉 = φ · 〈x, y〉 (x, y ∈ E). On the other hand, if F is a full Hilbert C∗-module over another commutative C∗-algebra C0(∆), we show that an “bi-orthogonality preserving” bijective map θ with some “local-type property” will be bounded and satisfy 〈θ(x), θ(y)〉 = φ · 〈x, y〉 ◦ σ (x, y ∈ E) for a map φ ∈ Cb(Ω)+ and a homeomorphism σ : ∆ → Ω. We will also have a look at the non-commutative situation.