A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis
Let script G sign be a locally compact group. Consider the Banach algebra L1 (script G sign)**, equipped with the first Arens multiplication, as well as the algebra LUC(script G sign)*, the dual of the space of bounded left uniformly continuous functions on script G sign, whose product extends the convolution in the measure algebra M(script G sign). 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  and Lau in , that the topological centres of the latter algebras precisely are L1 (script G sign) and M(script G sign), 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.
|Journal||Archiv der Mathematik|
Neufang, M. (2004). A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis. Archiv der Mathematik, 82(2), 164–171. doi:10.1007/s00013-003-0516-7