Assume $G$ is a locally compact abelian group and let $C^*(G)$ denote its group $C^*$-algebra.
I am reading a proof that uses the 'fact' that some $f\in C^*(G)$ is a limit of self-adjoint functions $f_n\in L^1(G)$.
Of course, since $L^1(G)$ is dense in its group $C^*$-algebra, $f$ can be written as a limit of integrable functions $f_n$, but why can we assume that $f_n=f_n^*$, i.e., $f_n(x)=\overline{f_n(x^{-1})}$? Does this require a special property of $f$?