On a conjecture by Ghahramani–Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A∗∗ equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A∗∗ is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani–Lau’s conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.) Furthermore, the same condition used above implies that every linear left A∗∗-module homomorphism on A∗ is automatically bounded and w∗-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A∗ A)∗, for A having a right approximate identity bounded by 1. © 2005 Elsevier Inc. All rights reserved.