On continuous representations in banach algebras

This thesis consists of six main chapters focusing on continuous representations in Banach algebras and some properties of Banach algebras. The first chapter provides general information about the thesis and a brief history of the topic. The second chapter presents a literature review. The third chapter introduces some basic definitions, theorems, and examples necessary for the subsequent chapters. Let G be a locally compact group and L^1 (G) be the group algebra defined on G. Let θ:L1(G)→A be a continuous homomorphism, where fi and hj are bounded approximate identities from the right and left, respectively, in L^1 (G). If L^1 (G) has a bounded approximate identity for █((f_i+h_j-f_i 〖*h〗_j ) #(1)) the relationship regarding the homomorphism █(θ(f_i+h_j-f_i*h_j ) #(2)) in A is shown to be a bounded approximate identity. Additionally, some results related to the homomorphism are presented. The fourth chapter examines some properties related to Arens products. It is shown that A^(**) also possesses a weak^* bounded approximate identity if A has both left and right weak^* bounded approximate identities. In the fifth chapter, the behavior of almost periodic functionals on Banach algebras was investigated. The relationships between the compactness and weak compactness properties of these functionals and Arens regularity were explored. In particular, necessary and sufficient conditions for the equality of the spaces ap(A) and wap(A) were established. The sixth and final chapter presents the conclusions and recommendations.