Minimal sets determining the topological centre of the algebra LUC(G)*

Abstract: The Banach algebra LUC(G)* associated to a topological group G has been of interest in abstract harmonic analysis. A number of authors have studied the topological centre of LUC(G), which is defined as the set of elements in LUC(G) for which the left multiplication is w*--w*-continuous on LUC(G). Several recent works show that for a locally compact group G it is sufficient to test the continuity of the left multiplication at just one specific point in order to determine whether an element of LUC(G) belongs to the topological centre. In this work we extend some of these results to a much larger class of groups which includes many non-locally compact groups as well as all the locally compact ones. This answers a question raised by H.G. Dales. We also obtain a corollary about the topological centre of any subsemigroup of LUC(G)* containing the uniform compactification of G. In particular, we prove that there are sets of just one point determining the topological centre of the uniform compactification itself.