Automatic continuity and $C_0(Ω)$-linearity of linear maps between $C_0(Ω)$-modules

Let $Ω$ be a locally compact Hausdorff space. We show that any local $\mathbb{C}$-linear map (where "local" is a weaker notion than $C_0(Ω)$-linearity) between Banach $C_0(Ω)$-modules are "nearly $C_0(Ω)$-linear" and "nearly bounded". As an application, a local $\mathbb{C}$-linear map $θ$ between Hilbert $C_0(Ω)$-modules is automatically $C_0(Ω)$-linear. If, in addition, $Ω$ contains no isolated point, then any $C_0(Ω)$-linear map between Hilbert $C_0(Ω)$-modules is automatically bounded. Another application is that if a sequence of maps $\{θ_n\}$ between two Banach spaces "preserve $c_0$-sequences" (or "preserve ultra-$c_0$-sequences"), then $θ_n$ is bounded for large enough $n$ and they have a common bound. Moreover, we will show that if $θ$ is a bijective "biseparating" linear map from a "full" essential Banach $C_0(Ω)$-module $E$ into a "full" Hilbert $C_0(Δ)$-module $F$ (where $Δ$ is another locally compact Hausdorff space), then $θ$ is "nearly bounded" (in fact, it is automatically bounded if $Δ$ or $Ω$ contains no isolated point) and there exists a homeomorphism $σ: Δ\rightarrow Ω$ such that $θ(e\cdot φ) = θ(e)\cdot φ\circ σ$ ($e\in E, φ\in C_0(Ω)$).

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