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 qu otient map G --> G/N induces a group algebra homomorphism T-N : l(1) (G) -- > l(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 approxi mate 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 l(1) (G), then T-N (I) is closed in l(1) (G/N). This answers a question of Reiter.