Let $G$ be a locally compact group and let $\Gamma$ be a closed subgroup of $G\times G$. In this paper, the concept of commutativity with respect to a closed subgroup of a product group, which is a generalization of multipliers under the usual sense, is introduced. As a consequence, we obtain characterization of operators on $L^2(G)$ which commute with left translation when $G$ is amenable.