Separating points by semicharacters in topological semigroups

Abstract

1. The semigroup analogue of the Pontrjagin duality theorem was first studied in [1]. In that paper, it was shown that a necessary and sufficient condition for duality in discrete abelian semigroups is that the semigroup be a union of groups and have an identity element. Such semigroups we shall call inverse semigroups. For compact abelian topological semigroups it was shown in [1] that the separation of points by semicharacters is a sufficient condition for duality in an inverse semigroup with identity. In [2] it was shown that, in any topological abelian semigroup, a necessary condition for duality is that the semigroup be an inverse semigroup with identity and continuous inversion. In this paper we obtain necessary and sufficient conditions that semicharacters separate points in a topological abelian inverse semigroup with identity which is compact, or locally compact with continuous inversion. In the compact case we obtain the same result as has been given by Sneperman [5], using different methods. 1.1. DEFINITION. An abelian semigroup is a nonempty set together with a map m: (x, y)-*xy on SXS to S, such that x(yz) = (xy)z and xy =yx for all x, y and z in S. If is a Hausdorff topological space and the mapping m is continuous, is called a topological abelian semigroup. 1.2. DEFINITION. A semicharacter X of a topological abelian semigroup is a bounded, continuous, complex-valued function on S, not identically zero, satisfying x(xy) =x(x)x(y) for all x and y in S. We denote the set of semicharacters of by SA. We endow SA with the compact open topology. The following facts aretobefoundin [1], [3]or [4]. 1.3. If has an identity element, S^ becomes a topological abelian semigroup, when endowed with the operation of pointwise multiplication. 1.4. If has an identity element, and is discrete, S^ is a compact abelian semigroup [1, 3.1 ]. 1.5. We call an abelian inverse semigroup if is an abelian semigroup which is a union of groups. If is a topological, abelian inverse semigroup with an identity element then S is of the same type. Further, if is compact then S^ is discrete [1, 6.11.