A famous factorization theorem of Cohen states that for any locally compact group $G$, $$L^1(G)=L^1(G)*L^1(G).$$ I want to know if analogous results exist for the class of smooth functions when $G$ is a Lie group. For instance, is it true that $$C^\infty(G)=C^\infty(G)*C^\infty(G)?$$

(Feel free to impose any simplifying assumption such as compactness of $G$, compactly supported functions, or $$C^\infty(G)=\text{Span } C^\infty(G)*C^\infty(G)$$ etc.)

It would be great if someone could point out some of the known positive or negative results or open problems that might exist. References are also highly appreciated.