Tietze extension theorem for Hilbert 𝐶*-modules

We prove the following generalization of the noncommutative Tietze extension theorem: if V V is a countably generated Hilbert C ∗ C^* -module over a σ \sigma -unital C ∗ C^* -algebra, then the canonical extension Φ ¯ \overline {\Phi } of a surjective morphism Φ : V → W \Phi : V \rightarrow W of Hilbert C ∗ C^* -modules to extended (multiplier) modules, Φ ¯ : V d → W d \overline {\Phi } : V_d \rightarrow W_d , is also surjective.