Radon–Nikodým property and Lau’s conjecture

There is a long-standing problem, posed by A.T.-M. Lau [Fixed point theory and its applications, Academic Press, New York-London, 1976, pp. 121–129], whether left amenability is sufficient to ensure the existence of a common fixed point for every jointly weak ∗ continuous nonexpansive semigroup action on a nonempty weak ∗ compact convex set in a dual Banach space. In this note we discuss the current status of this problem and give a partial solution in the case of weak ∗ compact convex sets with the Radon–Nikodým property.