On subspaces of invariant vectors

TL;DR

This paper investigates the existence and uniqueness of invariant subspace complements for group representations on Banach spaces, providing conditions for when such complements exist or do not, especially for actions on compact metric spaces.

Contribution

It establishes the non-existence of complemented invariant subspaces for certain non-amenable groups and identifies conditions for invariant complements in specific representations.

Findings

Non-amenable groups have representations with non-complemented fixed vector subspaces.

Conditions are identified for the existence of G-invariant complements.

Special focus on representations arising from group actions on compact metric spaces.

Abstract

Let $X_{\pi}$ be the subspace of fixed vectors for a uniformly bounded representation $\pi$ of a group $G$ on a Banach space $X$. We study the problem of the existence and uniqueness of a subspace $Y$ that complements $X_{\pi}$ in $X$. Similar questions for $G$-invariant complement to $X_{\pi}$ are considered. We prove that every non-amenable discrete group $G$ has a representation with non-complemented $X_{\pi}$ and find some conditions that provide an $G$-invariant complement. A special attention is given to representations on $C(K)$ that arise from an action of $G$ on a metric compact $K$.