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On subspaces of invariant vectors

Volume 236 / 2017

Tatiana Shulman Studia Mathematica 236 (2017), 1-11 MSC: 22A25, 46B99, 22D25. DOI: 10.4064/sm8378-11-2016 Published online: 25 November 2016


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 $K$.


  • Tatiana ShulmanInstitute of Mathematics
    Polish Academy of Sciences
    00-656 Warszawa, Poland

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