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Let $A$ be Banach algebra and $X,Y$ be two left Banach $A$-modules. It is said that the linear bounded map $\phi:X\to Y$ is left $A$-module homomorphism if for any $a\in A$ and any $x\in X$ we have $\phi(a.x)=a.\phi(x)$.

We define Local left $A$-module homomorphism. We say that $\phi:X\to Y$ is local left $A$-module homomorphism if for any $x\in X$ there exist a open neigborhood $V_x\subset X$ of $x$ and a left $A$-module homomorphism $\psi:X\to Y$ such that for all $y\in V_x$ we have $\phi(y)=\psi(y)$ i.e., $\phi=\psi$ on $V_x$.

First question: Is this definition well-defined? I mean is there a difference between local left $A$-module homomorphism and left $A$-module homomorphism?

Second question: Is this definition defined before?

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    $\begingroup$ Yes, both definition are the same. Since $\phi = \psi$ on a neighbbohood of $0$, then for $x \in X$ $a \in A$ and $t \in \mathbb{R}$ small enought, $\phi(ax) = t^{-1}.\phi(tax) = t^{-1}.\psi(tax) = \psi(ax) = a \psi(x) = a.t^{-1} \psi(tx) = a.t^{-1} \phi(tx) = a\phi(x)$. $\endgroup$
    – user10676
    Jun 18, 2014 at 14:05
  • $\begingroup$ wow!!!!!!!!!!!!!!!!!!!!!!!!!!!! $\endgroup$
    – user51514
    Jun 18, 2014 at 18:08

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On the first question: The difference between "general" and local here is that the general definition goes for "any a∈A and any x∈X", while the local version is only for a subspace "V⊂X".

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