irreducible representation of a group Reduced group $C^\ast$-algebra of group $G$ is defined to be $G^*_{r}(G)=\overline{\lambda(L^1(G))}$ where $\lambda$ is left regular representation. My question is how to get a irreducible representation of $G$ from a irreducible representation of reduced group $C^*$-algebra $G^*_{r}(G)$? (In my question $G$ is a locally compact group)

You need an approximate identity for the convolution, i.e., a Dirac delta function at the identity of $G$. Then you can get a Dirac-$\delta$ at each $g$ by translating. Your representations $g$ amounts to $\pi(\delta_g)$. Of course, this can formally only done with nets.
To obtain that Dirac delta for general locally compact groups can be done via approximation of $G$ locally as a projective limit of Lie groups.
