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.

**Added in Edit**: 

 1. While browing the web, I came across the [thesis][1] of Marc Palm, where he mentions on page 65 that

> Every smooth function is the convolution product of smooth functions,
> briefly denoted by $$C_c^\infty(G)=C_c^\infty(G)*C_c^\infty(G).$$

This is clearly stronger than the D-M result, and in contrast, in my opinion, to what Paul writes below in the comments: 

> A technical point is that finite sums are necessary in that context,
> so not every test function is exactly a single convolution of two.
Does Marc really mean $C_c^\infty(G)=\text{Span }C_c^\infty(G)*C_c^\infty(G)$?

 2. I also came across [this post][2] by Marc Palm where he gives an answer to my question below (in the comments).

  [1]: http://ediss.uni-goettingen.de/bitstream/handle/11858/00-1735-0000-000D-F074-7/palm.pdf?sequence=1
  [2]: https://mathoverflow.net/questions/124260/trace-class-functions-on-locally-compact-groups