Rigidity of commuting affine actions on reflexive Banach spaces

TL;DR

This paper proves that affine isometric actions of product groups on reflexive Banach spaces exhibit a rigidity property, either fixing a vector or almost fixing points, revealing structural constraints of such actions.

Contribution

It introduces a simple argument establishing a rigidity phenomenon for affine isometric actions on reflexive Banach spaces, particularly for product and abelian groups.

Findings

Either the linear part fixes a unit vector or the action almost fixes a point.

Affine isometric actions of abelian groups with no fixed vectors almost fix points.

The result applies to actions on reflexive Banach spaces, highlighting their structural properties.

Abstract

We give a simple argument to show that if {\alpha} is an affine isometric action of a product G x H of topological groups on a reflexive Banach space X with linear part {\pi}, then either {\pi}(H) fixes a unit vector or {\alpha}|G almost fixes a point on X. It follows that any affine isometric action of an abelian group on a reflexive Banach space X, whose linear part fixes no unit vectors, almost fixes points on X.