Replace the bounded-orbit hypothesis in the dual-space fixed-point characterization

Ascertain whether, in the dual Banach space setting of Theorem 3.5, the requirement that a separately continuous, equicontinuous, affine action of a semihypergroup K on X* admits at least one bounded orbit can be replaced by an equivalent, more conventional, and more easily verifiable condition on the representation without losing the fixed-point characterization of left amenability of AP(K).

Background

Theorem 3.5 establishes an equivalence between left amenability of AP(K) and the existence of common fixed points for certain actions on dual Banach spaces, provided at least one orbit is bounded. This bounded-orbit assumption is a technical hypothesis that guarantees compactness behavior necessary for fixed-point arguments.

The author explicitly notes that it is unknown whether this bounded-orbit condition can be replaced by an alternative, equivalent representation-theoretic assumption that is easier to check and more standard, leaving a gap for further refinement of the theorem’s hypotheses.