An extension of a non-commutative Choquet-Deny theorem

Let $G$ be a discrete group, and let $N$ be a normal subgroup of $G$. Then the quotient map $G\to G/N$ induces a group algebra homomorphism $T_N:\ell^1(G)\to \ell^1(G/N)$. It is shown that the kernel of this map may be decomposed as $\ker(T_N)=R+L$, where $R$ is a closed right ideal with a bounded left approximate identity and $L$ is a closed left ideal with a bounded right approximate identity. It follows from this fact that, if $I$ is a closed two-sided ideal in $\ell^1(G)$, then $T_N(I)$ is closed in $\ell^1(G/N)$. This answers a question of Reiter.