A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis

Research paper by M. Neufang

Indexed on: 01 Feb '04Published on: 01 Feb '04Published in: Archiv der Mathematik

Abstract

Let $$\cal G$$ be a locally compact group. Consider the Banach algebra $$L_{1}(\cal G)^{**}$$ , equipped with the first Arens multiplication, as well as the algebra LUC $$(\cal G)^*$$ , the dual of the space of bounded left uniformly continuous functions on $$\cal G$$ , whose product extends the convolution in the measure algebra M $$(\cal G)$$ . We present (for the most interesting case of a non-compact group) completely different - in particular, direct - proofs and even obtain sharpened versions of the results, first proved by Lau-Losert in [9] and Lau in [8], that the topological centres of the latter algebras precisely are $$L_{1}(\cal G)$$ and M $$(\cal G)$$ , respectively. The special interest of our new approach lies in the fact that it shows a fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, it avoids the use of any measure theoretical techniques. At the same time, deriving both results in perfect parallelity, our method reveals the nature of their close relation.