Invariant subspace theorems for amenable groups

Abstract

In [5], Ky Fan proved the following remarkable amenability “invariant subspace” theorem:Let G be an amenable group of continuous, invertible linear operators acting on a locally convex space E. Let H be a closed subspace of finite codimension n in E and X⊂E be such that:(i) H and X are G-invariant;(ii) (e + H) ∩X is compact convex for all e ∈ E;(iii) X contains an n-dimensional subspace V of E. Then there exists an n-dimensional subspace of E contained in X and invariant under G.