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