The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.

Let $G$ be a locally compact Abelian group and let $bG$ denote its Bohr compactification (the Pontryagin dual of $\widehat{G}$ with the discrete topology). Denote by $\mathfrak{A}$ the space $L_1(G)^{**}$ furnished with either Arens product.

Is there a canonical action of $M(bG)$ (the measure algebra on $bG$) on $L_\infty(G)$ that would give rise to an isometric homomorphism $M(bG)\to \mathfrak{A}$?