The study of the Banach algebra LUC ( G ) * associated to a topological group G has been of interest in abstract harmonic analysis. In particular, several authors have studied the topological centre Λ ( LUC ( G ) * ) of this algebra, which is defined as the set of elements μ ∈ LUC ( G ) * such that left multiplication by μ is w * − w * -continuous. In recent years, several works have appeared in which it is shown that for a locally compact group G it is sufficient to test the continuity of the left translation by μ at just one specific point in order to determine whether μ ∈ LUC ( G ) * belongs to Λ ( LUC ( G ) * ) . 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 Dales [Review of S. Ferri and M. Neufang, ‘On the topological centre of the algebra LUC ( G ) * for general topological groups’, J. Funct. Anal. 144 (2007) 154–171. Amer. Math. Soc. MathSciNet Mathematical Reviews, 2007]. We also obtain a corollary about the topological centre of any subsemigroup of LUC ( G ) * containing the uniform compactification G LUC of G. In particular, we shall prove that there are sets of just one point determining the topological centre of the uniform compactification G LUC itself.