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.
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