Factorization in finite-codimensional ideals of group algebras

Let G be a σ-compact, locally compact group and I be a closed 2-sided ideal with finite codimension in L¹(G). It is shown that there are a closed left ideal L having a right bounded approximate identity and a closed right ideal R having a left bounded approximate identity such that I = L + R. The proof uses ideas from the theory of boundaries of random walks on groups.