Weakly almost periodic Banach algebras on semi-groups

TL;DR

This paper characterizes when the second dual of a Banach algebra is a WAP-algebra, linking it to Arens regularity, and explores conditions under which measure algebras on semigroups are WAP-algebras.

Contribution

It establishes a characterization of WAP-algebras in terms of duality and Arens products, and relates measure algebras on semigroups to WAP properties.

Findings

Second duals of Banach algebras are WAP-algebras iff they are dual Banach algebras.

The second dual being a WAP-algebra is equivalent to the original algebra being Arens regular.

For semigroup measure algebras, WAP property of M_a(S) is equivalent to that of M_b(S).

Abstract

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 M_a(S) is a WAP-algebra if and only if the measure algebra M_b(S) is so.