Let $G$ be a locally compact group. Let $M(G)$ denote the measure algebra and $L^1(G)$ denote the group algebra on $G$. Then $M(G)$ acts on $L^1(G)$ by convolution. So by duality $M(G)$ acts on $L^1(G)^{*}$ via $$\langle f\cdot\mu,\phi\rangle=\langle f,\mu\star \phi\rangle,$$

where $\mu\in M(G)$, $f\in L^{\infty}(G)$ and $\phi\in L^1(G)$.

Again via duality we get an action of $M(G)$ on $L^1(G)^{**}$ $$\langle \mu \cdot n,f\rangle=\langle n,f\cdot\mu\rangle,$$ where $\mu\in M(G)$, $f\in L^{\infty}(G)$ and $n\in L^1(G)^{**}$.

Let $C_0(G)$ denote the continuous functions on $G$ vanishing at infinity.

I'm working on a problem and I can boil it down to the following question:

**Question.** Suppose that $n\in L^1(G)^{**}$ such that the mapping $$\psi_{n,f}:G\to\mathbb{C},\ \ x\mapsto \langle\delta_x \cdot n,f\rangle$$ belongs to $C_0(G)$, for all $f\in L^{\infty}(G)$. Can I infer any information regarding the element $n\in L^1(G)^{**}$? For what $n\in L^1(G)^{**}$ this can be true?