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## Studia Mathematica

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## Fixed point properties for semigroups on weak$^{*}$ closed sets of dual Banach spaces

### Tom 262 / 2022

Studia Mathematica 262 (2022), 1-21 MSC: Primary 47H10, 47H20; Secondary 47D03, 46B20. DOI: 10.4064/sm200613-31-3 Opublikowany online: 29 July 2021

#### Streszczenie

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.

#### Autorzy

• Anthony To-Ming LauDepartment of Mathematical
and Statistical Sciences
University of Alberta