Non-unitarisable representations and maximal symmetry

TL;DR

This paper explores the structure of non-unitarisable groups on Hilbert spaces, their maximal symmetry properties, and the implications for Banach space geometry, including strict convexity of transitive norms.

Contribution

It provides new structural insights into bounded non-unitarisable groups and their maximal symmetry, and connects these to rigidity results for automorphism groups of infinite trees.

Findings

Bounded non-unitarisable groups have a unique invariant complemented subspace.

Rigidity results restrict the form of bounded subgroups extending certain non-unitarisable representations.

Transitive norms on separable Banach spaces are necessarily strictly convex.

Abstract

We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of ${\rm Aut}(T)$ on $\ell_2(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of ${\rm GL}(\mathcal H)$ extending a well-known non-unitarisable representation of $\mathbb F_\infty$. As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex.