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.
