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

Abstract:

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.