On the topological centre of the algebra LUC∗(G) for general topological groups

We consider the Banach algebra LUC∗(G) for a not necessarily locally compact topological group G. Our goal is to characterize the topological centre Zt(LUC∗(G)) of LUC∗(G). For locally compact groups G, it is well known that Zt(LUC∗(G)) equals the measure algebra M(G). We shall prove that for every second countable (not precompact) group G, we have , where denotes the completion of G with respect to its right uniform structure (if G is precompact, then Zt(LUC∗(G))=LUC∗(G), of course). In fact, this will follow from our more general result stating that for any separable (or any precompact) group G, we have Zt(LUC∗(G))=Leb(G), where Leb(G) denotes the algebra of uniform measures. The latter result also partially answers a conjecture made by I. Csiszár 35 years ago [I. Csiszár, On the weak∗ continuity of convolution in a convolution algebra over an arbitrary topological group, Studia Sci. Math. Hungar. 6 (1971) 27–40]. We shall give similar results for the topological centre Λ(GLUC) of the LUC-compactification GLUC of G. In particular, we shall prove that for any second countable (not precompact) group G admitting a group completion, we have (if G is precompact, then Λ(GLUC)=GLUC). Finally, we shall show that every linear (left) LUC∗(G)-module map on LUC(G) is automatically continuous whenever G is, e.g., separable and not precompact.