Regularity of Segal algebras

Let A be a Banach algebra. The second dual A** can be equipped with two multiplications, each of which is a natural extension of the original multiplication in A. The algebra A is said to be Arens regular if these two multiplications coincide. We give necessary (and, for some classes of algebras, sufficient) conditions for the regularity of a Segal algebra. We also obtain necessary and sufficient conditions for the weak complete continuity of a Segal algebra.