Fixed point properties for semigroups on weak closed sets of dual Banach spaces
Volume 262 / 2022
Abstract
We study the long-standing problem as to when a left reversible semitopological semigroup S acting as nonexpansive mappings on a weak^{*} closed convex subset K of the dual space E^* of a Banach space E has a common fixed point in K. We show that, if the action is separately weak^{*} continuous and if there is an element b\in K such that the orbit Sb is bounded, then K has a common fixed point for S provided K has weak^{*} normal structure. If K is also L-embedded and Sb is weakly precompact, then K has a common fixed point for S provided K has weak normal structure. We also study the notion of local amenability on the space C_{\rm b}(S) of all bounded continuous functions on a semitopological semigroup S, concerning a problem posed by the first author in Marseille, 1989, which remains open now.
