Talin Budak, Nilgün Işık, John Pym; Minimal determinants of topological centres for some algebras associated with locally compact groups. Bull London Math Soc 2011; 43 (3): 495-506. doi: 10.1112/blms/bdq116
The second dual L1(G)** of the group algebra has a well-known Arens multiplication (μ, ν) ↦ μν which is weak* continuous in the μ variable. In contrast, if ν ↦ μν is weak* continuous at every point, then μ ∈ L1(G). A significant question is whether continuity at all points ν is necessary for this conclusion, and there has been a long-standing conjecture that continuity at just two specified points might be enough. The main conclusion of the present paper is that the minimal number is in fact just one. However, a second theorem considers a closely related question for which two points are required. The methods yield similar answers to corresponding problems about the quotient algebra LUC(G)* of L1(G)** and the compactification GLUC of the group G.