This thesis consists of two separate and distinct parts. Part One is concerned with the problem of characterizing of homomorphisms and derivations on the algebra L1(w). Chapter 1.1. is on general properties of L1(w). In this chapter we prove that every continuous endomorphism of L1(w) has an extension to a continuous endomorphism of M(w). In Chapter 1.2 we characterize isomorphisms from one semi-simple algebra L1(wl) onto another semi-simple algebra L'(W2). In this chapter we also study the endomorphisms of L1 (R+). In Chapter 1.3 we characterize the isometric isomorphisms of a radical L1 (w). We also find a necessary and sufficient condition for two radical algebras L1(w1) and L1(w2) to be isometrically isomorphic. Chapter 1.4 is on derivations of L1(w). In this chapter we characterize derivations on a radical L1 (w) and we find necessary and sufficient conditions on w for the existence of non-zero derivations. Part Two is on isometric representations of the algebras M(G). The main results of this part are in Chapter 2.2. In this chapter we prove that there is an isometric isomorphism from M(G) into BB(H) and the algebra L1 (G) is not isometrically isomorphic with an algebra of operators on a Hilbert space.