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

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.