On the topological centre of the algebra [formula omitted] for general topological groups

Abstract

We consider the Banach algebra LUC ( G ) ∗ for a not necessarily locally compact topological group G . Our goal is to characterize the topological centre Z t ( LUC ( G ) ∗ ) of LUC ( G ) ∗ . For locally compact groups G , it is well known that Z t ( LUC ( G ) ∗ ) equals the measure algebra M ( G ) . We shall prove that for every second countable (not precompact) group G , we have Z t ( LUC ( G ) ∗ ) = M ( G ˆ ) , where G ˆ denotes the completion of G with respect to its right uniform structure (if G is precompact, then Z t ( 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 Z t ( 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 ( G LUC ) of the LUC-compactification G LUC of G . In particular, we shall prove that for any second countable (not precompact) group G admitting a group completion, we have Λ ( G LUC ) = G ˆ (if G is precompact, then Λ ( G LUC ) = G LUC ). 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.