It is more common to just write $L_g$ for $L_{x_g}$. As $L(G)$ admits a finite trace, there is a natural injective map $L(G)$ into $\ell^2(G)$-- this is your map $A \mapsto (\mu_g)$. It is absolutely not true that this map surjects (Open Mapping Theorem). It is obviously sufficient that $(\mu_g)\in\ell^1(G)$ for there to be some $A$ giving rise to $(\mu_g)$.
With $G=\mathbb F_2$, one can say a bit more. For example, Haagerup showed in
Haagerup, Uffe
An example of a nonnuclear C∗-algebra, which has the metric approximation property.
Invent. Math. 50 (1978/79), no. 3, 279–293.
That if $f$ is a function supported on the collection of words of reduced length $n$, then there is $A\in L(G)$ induced $f$, with $\|A\| \leq (n+1) \|f\|_2$.