On topological groups with an approximate fixed point property.

Cleon S Barroso,Vladimir G Pestov,Brice R Mbombo

doi:10.1590/0001-3765201720150382

Abstract

A topological group G has the Approximate Fixed Point (AFP) property on a bounded convex subset C of a locally convex space if every continuous affine action of G on C admits a net ( x i ) , x i ∈ C , such that x i - g ⁢ x i ⟶ 0 for all g ∈ G . In this work, we study the relationship between this property and amenability.

Highlights

One of the most useful known characterizations of amenability is stated in terms of a fixed point property
An active branch of current research is devoted to the existence of approximate fixed points for single maps
We show that a discrete group G is amenable if and only if every continuous affine action of G on a bounded convex subset C of a locally convex space (LCS) X admits approximate fixed points

Summary

Introduction

One of the most useful known characterizations of amenability is stated in terms of a fixed point property. We show that a discrete group G is amenable if and only if every continuous affine action of G on a bounded convex subset C of a locally convex space (LCS) X admits approximate fixed points. A Polish group G is amenable if and only if every continuous affine action of G on a convex, compact and metrizable subset K of a locally convex space X admits a fixed point.

Results

Conclusion