Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

Abstract

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map (a, b) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC (G) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f ε C(G) such that the map a → laf of G into C(G) is continuous when C(G) has a norm topology, where (laf )(x) = f (ax) (a, x ε G). Then LUC (G) is a closed subalgebra of C(G) invariant under translations. Furthermore, if m ε LUC (G)*, f ε LUC (G), then the functionis also in LUC (G). Hence we may define a productfor n, m ε LUC(G)*. LUC (G)* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l∞(G)* = l1(G)** to LUC (G)*. We refer the reader to [1] and [10] for more details.