Arens regularity of banach algebras

This thesis consists of five main chapters. In the first chapter, the introduction, the historical development of the subject is explained, and general information about the thesis content and chapters is provided. The second chapter presents the fundamental definitions and concepts necessary for the third, fourth, and fifth chapters. The third chapter comprises two subsections. This chapter deals with the Arens regularity of a Banach algebra. Its definition is given, and examples are provided. In particular, characterizations for a Banach algebra to be Arens regular are presented, and additionally, it is shown that any closed subalgebra of an Arens regular Banach algebra is also Arens regular. Chapter Four consists of five subsections. In this chapter, the concept of weakly Arens regular Banach algebras is defined, and the notion of quasi-multipliers is extended to a special class of Banach algebras. Among other topics, the relationship between the Arens regularity of a weakly Arens regular Banach algebra A and the space QM_r (A^*) , which consists of all bilinear and separately continuous right quasi-multipliers of A^*, is considered. Definitions, theorems, and examples related to these topics are presented. Additionally, various properties of the strict and quasi-strict topologies QM_r (A^*) are provided. In Chapter Five, an introduction to the theory of semi-multipliers on Banach algebras with approximate identities is presented, along with relevant definitions and theorems. Furthermore, applications to C^*-algebras and group algebras are developed and enriched with examples. Sixth, the chapter presents the conclusions and recommendations.