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. 



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> 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 $M(bG)\to \mathfrak{A}$?
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Measure algebra on the Bohr compactification vs the bidual algebras
I'm going to say no. The "canonical" pairing of $M(bG)$ with $L^\infty(G)$ is to integrate a function in $L^\infty(G)$ against the restriction to $G$ of a measure in $M(bG)$. But this is not faithful: any measure supported on $bG\setminus G$ would go to zero in $L^\infty(G)^\*$. To pick up mass on this corona we would want to extend functions in $L^\infty(G)$ to $bG$. But you have to be almost periodic to canonically extend to $bG$, so it doesn't seem like there's any way to get what you want.


