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

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.

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