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NON-UNITARISABLE REPRESENTATIONS AND MAXIMAL SYMMETRY
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 2 / April 2017
- Published online by Cambridge University Press:
- 01 July 2015, pp. 421-445
- Print publication:
- April 2017
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- Article
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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
$\text{Aut}(T)$ on 
$\ell _{2}(T)$, where 
$T$ is the countably infinite regular tree, to describe the possible bounded subgroups of 
$\text{GL}({\mathcal{H}})$ 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.
