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 → G/N induces a group algebra homomorphism T_N : ℓ¹(G) → ℓ¹(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 ℓ¹(G), then T_N(I) is closed in ℓ¹(G/N). This answers a question of Reiter.