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

Let $\mathcal{G}$ 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 $L_1(\mathcal{G}, w)$ is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of $L_1(\mathcal{G}, w)^{**}$ equal $L_1(\mathcal{G}, w)$. Also, we show that the topological centre of the algebra $\mathrm{LUC} \left(\mathcal{G}, w^{-1} \right)^*$ equals the weighted measure algebra $\mathrm{M}(\mathcal{G} , w)$. Moreover, still in the same situation, we prove that every linear (left) $L_\infty(\mathcal{G}, w^{-1})^{*}$-module map on $L_\infty \left(\mathcal{G}, w^{-1} \right)$ is automatically bounded, and even $w^{*}$-$w^{*}$-continuous, hence given by convolution with an element in $\mathrm{M}(\mathcal{G},w)$. To this end, we derive a general factorization theorem for bounded families in the $L_\infty \left(\mathcal{G} , w^{-1} \right)^*$-module $L_\infty \left(\mathcal{G}, w^{-1} \right)$. Finally, using this result in the case where $w \equiv 1$, we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup $\mathcal{G}^{\mathrm{LUC}} \setminus \mathcal{G}$ is empty, where $\mathcal{G}^{\mathrm{LUC}}$ denotes the $\mathrm{LUC}$-compactification of $\mathcal{G}$. 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.