Let $G$ be a unimodular locally compact group (my main examples are algebraic groups over local fields. Thefore we can assume $G$ is Type I, if necessary). Then there are at least three group algebras associated to $G$ : the algebra $C_c(G)$ (with convolution) of continuous functions of compact support, the algebra $L^1(G)$ (with convolution), and the $C^\star$-algebra of $G$, denoted by $C^\star(G)$.

Let $I$ be a primitive ideal in $C^*(G)$.

(a) Is it true that $I\cap L^1(G)$ is dense in $I$?

(b) Is it true that $I\cap C_c(G)$ is dense in $I$? If not, then is it true that $I\cap C(G)$ is dense in $I$? (Here $C(G)$ denotes the space of continuous functions on $G$.