In this thesis, we present an expanded account of the second dual of Banach algebras with a continuous involution. Let A be an arbitrary Banach algebra and be the second dual of A with first Arens product. We discuss whether there is an involution on A extending the involution of A; in particular, we consider the second dual of group algebra related of locally compact group as . and , Also we characterize all isometric involution on L(G) and M(G). Lastly, we will discuss the relationship between the representation and positive definite functions.