In this paper we define the notion of weak Arens regular Banach algebras and extend the concept of quasi-multipliers to this certain class of Banach algebras. Among other the relationship between Arens regularity of the algebra A * * of a weak Arens regular Banach algebra A and the space QM r (A * ) of all bilinear and separately continuous right quasimultipliers of A * is investigated. Further, we stablish several properties of the strict and quasi-strict topologies on the QM r (A * )
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