On subspaces of invariant vectors

Tatiana Shulman ¹

¹ Institute of Mathematics Polish Academy of Sciences 00-656 Warszawa, Poland

Studia Mathematica, Tome 236 (2017) no. 1, pp. 1-11

Résumé

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 a $G$-invariant complement. A special attention is given to representations on $C(K)$ that arise from an action of $G$ on a metric compact $K$.

Détail

DOI : 10.4064/sm8378-11-2016

Keywords: subspace fixed vectors uniformly bounded representation group banach space study problem existence uniqueness subspace complements similar questions g invariant complement considered prove every non amenable discrete group has representation non complemented conditions provide g invariant complement special attention given representations arise action metric compact