Since you are in the commutative setting, you can present the construction more simply. $M(G)=L_1(G)\oplus_1 S(G)$, where $S(G)$ is the space of complex measures that are singular with respect to Haar measure, and hence $M(G)^∗=L_\infty (G)\oplus_1 S(G)^∗$.

Since you are in the commutative setting, you can present the construction more simply. $M(G)=L_1(G)\oplus_1 S(G)$, where $S(G)$ is the space of complex measures that are singular with respect to Haar measure, and hence $M(G)^∗=L_\infty (G)\oplus_\infty S(G)^∗$.