  Let G be a locally compact group. Consider the Banach algebra L_1(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 L_1(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.
