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 })