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

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.