A Unified Approach to the Topological Centre Problem for Certain Banach Algebras Arising in Abstract Harmonic Analysis

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Let G be a locally compact group. Consider the Banach algebra L₁(G)**, equipped with the first Arens multiplication, as well as the algebra LUC(G)*, the dual of the space of bounded left uniformly continuous functions on G, whose product extends the convolution in the measure algebra M(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 and Lau, that the topological centres of the latter algebras precisely are L₁(G) and M(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.