THE EXISTENCE OF NONZERO COMPACT RIGHT MULTIPLIERS AND ARENS REGULARITY OF WEIGHTED BANACH ALGEBRAS

Let omega be a weight function on a locally compact group g and let M(g, omega)*0 be the subspace of M(g, omega)* consisting of all functionals that vanish at infinity. We first introduce an Arens product on (M(g, omega)*0)* under which it is a Banach algebra. We then show that the existence of a nonzero compact right multiplier on (M(g, omega)*0)* is equivalent to compactness of g. We also prove that if (M(g, omega )*0)* is Arens regular, then g is discrete.