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

Author: Matthias Neufang
Journal: Proc. Amer. Math. Soc. 136 (2008), 1831-1839
MSC (2000): Primary 22D15, 43A10, 43A20, 43A22, 46H40, 54D35
Published electronically: January 30, 2008
MathSciNet review: 2373615
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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.

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Additional Information

Matthias Neufang
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6 Canada

Keywords: Locally compact group, weighted group algebra, left uniformly continuous function, Arens product, topological centre, semigroup compactification.
Received by editor(s): June 26, 2006
Received by editor(s) in revised form: August 31, 2006
Published electronically: January 30, 2008
Additional Notes: The present work was partly supported by NSERC. This support is gratefully acknowledged.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society

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