Let \(G\) be a separable locally compact group with fixed left Haar measure and let \(M(G)\) be the measure algebra on \(G\). For each probability measure \(\mu\) in \(M(G)\), let \(J_\mu= \{f- f* \mu\}^-\). Then \(J_\mu\) is a closed left ideal in \(L^1(G)\) and the collection \(J(G)\) of \(J_\mu\) for a probability measure \(\mu\) in \(M(G)\) is partially ordered by set inclusion. The author proves in this paper that if either \(J(G)\), or the collection \(J_a(G)\) of \(J_\mu\) for an absolutely continuous probability measure \(\mu\) in \(M(G)\), or the collection \(J_d(G)\) of \(J_\mu\) for a discrete probability measure \(\mu\) in \(M(G)\), has a unique maximal element, then \(G\) is amenable. This answers a problem of G. A. Willis. Conversely, if, in addition, \(G\) is connected and not amenable, then both \(J_a(G)\) and \(J_d(G)\) have infinitely many maximal elements.