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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.