Approximately orthogonality preserving mappings on Hilbert \(C_{0}(Z)\)-modules

In this paper, we will use the categorical approach to Hilbert \(C^{\ast}\)-modules over a commutative \(C^{\ast}\)-algebra to investigate the approximately orthogonality preserving mappings on Hilbert \(C^{\ast}\)-modules over a commutative \(C^{\ast}\)-algebra. Indeed, we show that if \(\Psi:\Gamma \rightarrow \Gamma^{\prime} \) is a nonzero \( C_{0}(Z) \)-linear \(( \delta , \varepsilon)\)-orthogonality preserving mapping between the continuous fields of Hilbert spaces on a locally compact Hausdorff space \(Z\), then \(\Psi\) is injective, continuous and also for every \( x, y \in \Gamma \) and \(z \in Z\), \[ \vert \langle \Psi(x),\Psi(y) \rangle(z) - \varphi^2(z) \langle x,y \rangle(z) \vert \leq \frac{4(\varepsilon - \delta)}{(1-\delta)(1+\varepsilon)} \Vert \Psi(x) \Vert \Vert \Psi(y) \Vert, \] where \(\varphi(z) = \sup \{ \Vert \Psi(u)(z) \Vert : u ~ \text{is a unit vector in} ~ \Gamma \}\).