Weak⁎ fixed point property and asymptotic centre for the Fourier–Stieltjes algebra of a locally compact group

In this paper we show that the Fourier–Stieltjes algebra B(G) of a non-compact locally compact group G cannot have the weak⁎ fixed point property for nonexpansive mappings. This answers two open problems posed at a conference in Marseille-Luminy in 1989. We also show that a locally compact group is compact exactly if the asymptotic centre of any non-empty weak⁎ closed bounded convex subset C in B(G) with respect to a decreasing net of bounded subsets is a non-empty norm compact subset. In particular, when G is compact, B(G) has the weak⁎ fixed point property for left reversible semigroups. This generalizes a classical result of T.C. Lim for the circle group. As a consequence of our main results we obtain that a number of properties, some of which were known to hold for compact groups, in fact characterize compact groups.