Invariant Subspaces for Algebras of Linear Operators and Amenable Locally Compact Groups

Abstract

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