Let be G a locally compact, non-compact group (we make the non- compactness assumption, for the most part, simply to avoid trivialities). We show that under a very mild assumption on the weight function w,the weighted group algebra L1(G, ω) is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of L1(G, ω)* equal L1(G, ω). Also, we show that the topological centre of the algebra LUC (G, ω-1) equals the weighted measure algebra M(G, ω). Moreover, still in the same situation, we prove that every linear (left) L∞(G, ω-1)*-module map on L∞ (G, ω-1) is automatically bounded, and even ω* -ωz,ast; -continuous, hence given by convolution with an element in M(G,ω) To this end, we derive a general factorization theorem for bounded families in the L∞ (G,w-1) -module L∞ (G, ω - 1). Finally, using this result in the case where ω ≡1, we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup GLUC \G is empty, where G LUC denotes the LUC-compactification of G.This sharpens an earlier result by Lau and Pym; moreover, our method of proof gives a partial answer to a problem raised by Lau and Pym in 1995.

Proceedings of the American Mathematical Society
School of Mathematics and Statistics

Neufang, M. (2008). On the topological centre problem for weighted convolution algebras and semigroup compactifications. Proceedings of the American Mathematical Society, 136(5), 1831–1839. doi:10.1090/S0002-9939-08-08908-9