Weakly almost periodic Banach algebras on semi-groups

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Let WAP(A) be the space of all weakly almost periodic functionals on a Banach algebra A. The Banach algebra A for which the natural embedding of A into WAP(A)* is bounded below is called a WAP-algebra. We show that the second dual of a Banach algebra A is a WAP-algebra, under each Arens products, if and only if A** is a dual Banach algebra. This is equivalent to the Arens regularity of A. For a locally compact foundation semigroup S, we show that the absolutely continuous semigroup measure algebra Mₐ(S) is a WAP-algebra if and only if the measure algebra Mb(S) is so.