Consider a pair $(G,phi)$ where $G$ is a (discrete) Abelian group and $phicolon Gto G$ is an endomorphism of $G$. There is a usual trick to construct a new pair $(G’,phi’)$ with the property that $phi’$ is an automorphism, and such that $(G’,phi’)$ is, in a certain sense, “as close as possible” to $(G,phi)$. Indeed, consider the following direct system:
$$
Goverset{phi}{longrightarrow}Goverset{phi}{longrightarrow}Goverset{phi}{longrightarrow}Goverset{phi}{longrightarrow}cdots
$$
Then one defines $G’$ as the colimit of the above system and lets $phi’$ be the map induced by $phi$.
Another, equivalent, approach is to see the pair $(G,phi)$ as a $mathbb Z(X)$-module $G_phi$ (where multiplication by $X$ corresponds to an application of $phi$), then $(G’,phi’)$ is the pair that corresponds with the $mathbb Z(X)$-module
$$G’_{phi’}:=G_{phi}otimes_{mathbb Z(X)}mathbb Z(X^{pm 1})$$
obtained by tensoring with $mathbb Z(X^{pm1})$ (the ring of Laurent polynomials). In this language it is easy to see that the obvious map $varphicolon G_{phi}to G’_{phi’}$ satisfies a universal property, in that any morphism of $mathbb Z(X)$-modules from $G_phi$ to a $mathbb Z(X)$-module $H_psi$ with $psi$ invertible, factors uniquely through $varphi$ (this makes the “as close as possible” in my second sentence precise).
Now the question is, What if the $G$ is a locally compact Abelian (always Hausdorff) topological group, and $phi$ is a continuous endomorphism? Can we do something similar to the above in this more general situation?
If it helps, I am happy to make some assumptions about $phi$, but I am looking for a construction that works when $phi$ is not necessarily open.
To conclude, let me precise that I am looking for functorial constructions that take as input the pair $(G,phi)$ (with $G$ LCA and $phi$ continuous), and give as output a pair $(G’,phi’)$ with $G’$ Hausdorff topological group, $phi’$ a topological automorphism (i.e., continuous, open and linear), that satisfy some kind of universal property as the one above.