Arens multiplication on Banach algebras related to locally compact semigroups

✍ Scribed by S. Maghsoudi; R. Nasr–Isfahani; A. Rejali

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 184 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200710691

✦ Synopsis

Let S be a locally compact semigroup, let ω be a weight function on S, and let Ma(S, ω) be the weighted semigroup algebra of S. Let L ∞ 0 (S; Ma(S, ω)) be the C * -algebra of all Ma(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; Ma(S, ω)) * under which Ma(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; Ma(S, ω)) * . We then study Arens regularity of L ∞ 0 (S; Ma(S, ω)) * and its relation with Arens regularity of Ma(S, ω), M (S, ω) and the discrete convolution algebra 1 (S, ω). As the main result, we prove that L ∞ 0 (S; Ma(S, ω)) * is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster.