Compact elements in weighted discrete semigroup algebras

Let S be a semigroup, and let omega be a weight on S. Then the Banach space l(1) (S, omega) is a Banach algebra with respect to the usual convolution product. We investigate the elements of l(1) (S, omega) which act compactly. The following is the main result of this paper. Let S be a non-unital, countable, cancellative, torsion-free, abelian semigroup; in particular, let S be a non-unital, countable subsemigroup of R+. Then there exists a weight omega on S such that every element of l(1) (S, omega) acts compactly.