On the topological centre problem for weighted convolution algebras and semigroup compactifications

Let G be 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 L-1(G, w) is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of L-1(G, w)** equal L-1(G, w). Also, we show that the topological centre of the algebra LUC (G, w(-1))* equals the weighted measure algebra M(G, w). Moreover, still in the same situation, we prove that every linear (left) L-infinity(G, w(-1))*-module map on L-infinity (G, w(-1)) is automatically bounded, and even w*-w*-continuous, hence given by convolution with an element in M(G, w). To this end, we derive a general factorization theorem for bounded families in the L-infinity (G, w(-1))(*)-module L-infinity (G, w(-1)). Finally, using this result in the case where w equivalent to 1, we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup G(LUC) \ 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.