Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the convolution: $$(F \ast G)(h) = \sum_{g \in G} F(g)G(hg^{-1})$$ When does an element of this algebra have a left inverse? Is this algebra well-studied? I am searching for references on these algebras.

P.S. $\lim_{g \to \infty} |F(g)|_p = 0 $ simply means that for every $\varepsilon > 0$, there is a finite subset of $G$ such that outside of this subset, $|F(g)|_p < \varepsilon$.