Invariant subspaces for algebras of linear operators and amenable locally compact groups

Abstract

Let G G be a locally compact group. We prove in this paper that G G is amenable if and only if the group algebra L 1 ( G ) {L_1}\left ( G \right ) (respectively the measure algebra M ( G ) M\left ( G \right ) ) satisfies a finite-dimensional invariant subspace property T ( n ) T\left ( n \right ) for n n -dimensional subspaces contained in a subset X X of a separated locally convex space E E when L 1 ( G ) {L_1}\left ( G \right ) (respectively M ( G ) M\left ( G \right ) ) is represented as continuous linear operators on E E . We also prove that for any locally compact group, the Fourier algebra A ( G ) A\left ( G \right ) and the Fourier Stieltjes algebra B ( G ) B\left ( G \right ) always satisfy T ( n ) T\left ( n \right ) for each n = 1 , 2 , … n = 1,2, \ldots .