Let $G$ be a topological group, $\mathcal{U}_e$ a neighborhood of identity $e$ (we can suppose $\mathcal{U}_e^{-1} \subseteq \mathcal{U}_e$), $V$ a $K-$vector space, $\rho: \mathcal{U}_e \to GL(V)$ such that $\rho(e) = \mathrm{id}_V$ and $\rho(g1) \rho(g2) = \rho(g_1 g_2)$ if $g_1 g_2 \in \mathcal{U}_e$ ($\rho$ is a local representation of $G$). I'm trying to extend $\rho$ to a representation of the whole group if $G$ is path connected and simply connected.
The ideia is simple: since $G$ is connected, for all $g \in G$ there are $g_1. g_2, \ldots, g_n \in \mathcal{U}_e$ such that $g = g_1 \ldots g_n$ and we define $\widehat{\rho}(g) = \rho(g_1) \ldots \rho(g_n)$.
Now, the question: is it well defined?
A priori, we have a set $\widehat{\rho}(g) = \{ \rho(g_1) \ldots \rho(g_n) \in GL(V) \mid g_1, \ldots, g_n \in \mathcal{U}_e, \quad g_1 \ldots g_n = g\}$. I guess that, if I can prove this set is discrete, by defining some paths and use the simply connected property it will be ok. Actually, I guess it's enough to prove $\widehat{\rho}(e)$ is discrete. I know that $\widehat{\rho}(e) < GL(V)$, so, i just need to prove that $\{\mathrm{id}_V\}$ is open.
If it's the case, we can generalize to any local homomorphism $\rho: \mathcal{U}_e \to H$ ?