LINEAR ORTHOGONALITY PRESERVERS OF HILBERT BUNDLES

Abstract

AbstractA ℂ-linear map θ (not necessarily 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 an orthogonality preserving map from a full Hilbert C0(Ω)-module E into another Hilbert C0(Ω) -module F that satisfies a weaker notion of C0 (Ω) -linearity (called ‘localness’), then θ is bounded and there exists ϕ∈Cb (Ω)+ such that 〈θ(x),θ(y)〉=ϕ⋅〈x,y〉 for all x,y∈E.