Extend the invariant-renorming characterization to locally compact groups

Establish an analogue for locally compact groups of the equivalence between the strongly Choquet–Deny property and the existence of equivalent invariant strictly convex norms for separable Banach-space representations.

Background

The main representation-theoretic characterization proved in the paper states that a countable group is strongly Choquet–Deny if and only if every uniformly bounded representation on a separable Banach space admits an equivalent invariant strictly convex norm. The definition of strongly Choquet–Deny groups itself can also be formulated for locally compact groups.

The paper does not establish the corresponding equivalence in the locally compact setting. It does prove a related positive result for topological groups having a closed abelian subgroup with compact quotient, but this does not provide a general locally compact analogue of the main theorem.