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The Banach algebra LUC(G)* associated to a topological group G has been ofinterest in abstract harmonic analysis. A number of authors have studied thetopological centre of LUC(G)*, which is defined as the set of elements inLUC(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 sufficientto test the continuity of the left multiplication at just one specific point inorder to determine whether an element of LUC(G)* belongs to the topologicalcentre. In this work we extend some of these results to a much larger class ofgroups which includes many non-locally compact groups as well as all thelocally compact ones. This answers a question raised by H.G. Dales. We alsoobtain a corollary about the topological centre of any subsemigroup of LUC(G)*containing the uniform compactification of G. In particular, we prove thatthere are sets of just one point determining the topological centre of theuniform compactification itself.

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