Abstract: In this lecture we first give some examples of states and GNS constructions. Then we introduce the reduced group $C^*$-algebra of a discrete group. Moreover, the Hecke pair $(G,H)$ consisting of a group $G$ and an almost normal subgroup $H$ will be introduced. Indeed, a subgroup $H$ of a group $G$ is called almost normal if every double coset $HgH$ can be written as a union of finite number of left cosets. Using Hecke pairs we introduce Hecke $C^*$-algebras as a generalization of group $C^*$-algebras. Finally we piont to the Bost-Connes pair and its Hecke $C^*$-algebra.