Zero divisors and $L^p(G)$

Let $G$ be a discrete group, $\mathbb{C}G$ the group ring of $G$ over $\mathbb{C}$ and $L^p(G)$ the Lebesgue space of $G$ with respect to Haar measure. It is known that if $G$ is torsion free elementary amenable, $0\ne \alpha\in \mathbb{C}G$ and $0\ne \beta\in L^2(G)$, then $\alpha*\beta\ne 0$. We will give a sufficient condition for this to be true when $p>2$, and in the case $G=\mathbb{Z}^n$ we will give sufficient conditions for this to be false when $p>2$.