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NON-UNITARISABLE REPRESENTATIONS AND MAXIMAL SYMMETRY

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  01 July 2015

Valentin Ferenczi  and
Christian Rosendal
Valentin Ferenczi
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão 1010, Cidade Universitária, 05508-90 São Paulo, SP, Brazil Equipe d’Analyse Fonctionnelle, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France (ferenczi@ime.usp.br)
Christian Rosendal
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA (rosendal.math@gmail.com) URL: http://homepages.math.uic.edu/∼rosendal
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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 $\text{Aut}(T)$ 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.

Keywords

unitarisable representationsmaximal norms

MSC classification

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B04: Isometric theory of Banach spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 2 , April 2017 , pp. 421 - 445
Copyright
© Cambridge University Press 2015 

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