Given a locally compact group $G$ let $\mathcal{J}_a(G)$
denote the set of all closed left ideals $J$ in $L^1(G)$ which
have the form $J=[L^1(G)*(\delta_e
-\mu)]\overline{\vphantom{t}\ }$ where $\mu$ is an absolutely
continuous probability measure on $G$. We explore the order
structure of $\mathcal{J}_a(G)$ when $\mathcal{J}_a(G)$ is
ordered by inclusion. When $G$ is connected and amenable we
prove that every nonempty family $\mathcal{F}\subseteq
\mathcal{J}_a(G)$ admits both a minimal and a maximal element;
in particular, every ideal in $\mathcal{J}_a(G)$ contains an
ideal that is minimal in $\mathcal{J}_a(G)$. Furthermore, we
obtain that every chain in $\mathcal{J}_a(G)$ is necessarily
finite. A natural generalization of these results to almost
connected amenable groups is discussed. Our proofs use results
from the theory of boundaries of random walks.