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

作者：Fendler Gero^*; Lau Anthony To Ming; Leinert Michael

来源：Journal of Functional Analysis, 2013, 264(1): 288-302.

DOI：10.1016/j.jfa.2012.10.011

• Weak-fixed point property
• Fourier-Stieltjes algebra
• Compact groups
• Asymptotic centre
• Left reversible semigroups
• Fell 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.

• 出版日期2013-1-1