@paulgarrett I was looking at the same essay:) but Casselman doesn't mention any "symmetric" factorization $g*g^*$ theorem there. Perhaps it is not to hard to see, but not very clear to me.

@paulgarrett D-M should be a strong positive result for many practical purposes. I just wanted to know what work has been done in this direction. As a separate question, does this follow from D-M that any function $f\in C_c^\infty(G)$ is a linear combination of the convolutions of the form $g*g^*$ in $C_c^\infty(G)*C_c^\infty(G)$? I think this is true but I don't have a reference for it. I really appreciate your help and clarification.

@paulgarrett You are absolutely right, D-M result is completely relevant to my question. My interpretation of D-M in this context is that any function in $C_c^\infty(G)$ can be approximated by a linear combination of the functions in $C_c^\infty(G)*C_c^\infty(G)$. Now I have two questions. (1) is this the right conclusion from D-M? (2) if the answer to the previous question is yes, then isn't the same conclusion clear from the fact that any function in $f\in C_c^\infty(G)$ can be approximated by $g_t*f$, where $g_t$ is an approximate unit? Sorry if these are too naive questions.

@paulgarrett The version of Dixmier-Malliavin that I have in mind is (roughly) the following: "Every smooth vector in a Frechet representation $(\pi, V)$ belongs to the Garding space."