Corollary

— Let $W$ be a closed linear subspace of $L^1(G, \beta)$. For $W$ to be a left (resp. right) ideal of $L^1(G, \beta)$, it is necessary and sufficient that $W$ be invariant under the left (resp. right) translations of $G$.

Suppose that $W$ is a left ideal. Let $s \in G$ and $g \in W$. We have $\varepsilon_s * g = \lim_V f_V * (\varepsilon_s * g) = \lim_V (f_V * \varepsilon_s) * g$, and $(f_V * \varepsilon_s) * g \in W$, therefore $\varepsilon_s * g \in W$, thus $\gamma(s)g \in W$. Conversely, if $W$ is invariant under the left translations, then $\mu *^\beta g \in W$ for $\mu \in \mathscr{M}^1(G)$ and $g \in W$, therefore $W$ is a fortiori a left ideal of $L^1(G, \beta)$. One argues similarly for right ideals.