20080501
On the topological centre problem for weighted convolution algebras and semigroup compactifications
Publication
Publication
Proceedings of the American Mathematical Society , Volume 136  Issue 5 p. 1831 1839
Let be G a locally compact, noncompact 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, ω1) equals the weighted measure algebra M(G, ω). Moreover, still in the same situation, we prove that every linear (left) L∞(G, ω1)*module map on L∞ (G, ω1) 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,w1) 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 LUCcompactification 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.
Additional Metadata  

doi.org/10.1090/S0002993908089089  
Proceedings of the American Mathematical Society  
Organisation  School of Mathematics and Statistics 
Neufang, M. (2008). On the topological centre problem for weighted convolution algebras and semigroup compactifications. Proceedings of the American Mathematical Society, 136(5), 1831–1839. doi:10.1090/S0002993908089089
