Zero divisors and L^p(G), II

Let G be a discrete group, let $p \ge 1$, and let $L^p(G)$ denote the Banach space $\{\sum_{g\in G} a_g g \mid \sum_{g\in G} |a_g|^p < \infty\}$. The following problem will be studied: given $0 \ne \alpha \in CG$ and $0 \ne \beta \in L^p(G)$, is $\alpha * \beta \ne 0$? We will concentrate on the case G is a free abelian or free group.