Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S , and let M a ( S , ω ) be the weighted semigroup algebra of S . Let L ∞ 0 ( S ; M a ( S , ω )) be the C * ‐algebra of all M a ( S , ω )‐measurable functions g on S such that g / ω vanishes at infinity. We introduce and study an Arens multiplication on L ∞ 0 ( S ; M a ( S , ω )) * under which M a ( S , ω ) is a closed ideal. We show that the weighted measure algebra M ( S , ω ) plays an important role in the structure of L ∞ 0 ( S ; M a ( S , ω )) * . We then study Arens regularity of L ∞ 0 ( S ; M a ( S , ω )) * and ist relation with Arens regularity of M a ( S , ω ), M ( S , ω ) and the discrete convolution algebra ℓ 1 ( S , ω ). As the main result, we prove that L ∞ 0 ( S ; M a ( S , ω )) * is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)