Characterize semitopological semihypergroups with common fixed points for weak*-continuous nonexpansive actions

Characterize coset spaces of locally compact groups and, more generally, semitopological semihypergroups that admit common fixed points for all weak*-continuous, norm-nonexpansive representations on nonempty weak*-compact convex subsets of a locally convex space or on the dual of a Banach space.

Background

The paper studies fixed-point properties of semihypergroup representations and their relationship with amenability of the space AP(K) of almost periodic functions. While many classical results exist for semigroups and groups, the broader category of semihypergroups introduces additional analytic subtleties.

In the introduction, the author highlights a long-standing open problem concerning a structural characterization of coset spaces and general semitopological semihypergroups that have common fixed points under weak*-continuous, norm-nonexpansive actions on weak*-compact convex sets in dual settings. The paper advances related characterizations via amenability of AP(K), but does not claim to fully resolve this classification problem.