The Dixmier problem, lamplighters and Burnside groups

Abstract

J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.

4 pages

References in corpus (2)

• Non-unitarisable representations and random forests
• $L^2$-Betti numbers and non-unitarizable groups without free subgroups

Cited by in corpus (10)

• Groups of piecewise projective homeomorphisms
• Expanders have a spanning Lipschitz subgraph with large girth
• On isometry groups and maximal symmetry
• A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra
• Group cubization (with an appendix by Mikael Pichot)
• $C^{\ast }$-simple groups without free subgroups
• Amenable groups and a geometric view on unitarisability
• The $M_d$-Approximation Property and Unitarisability
• Future directions in locally compact groups
• On the Dixmier problem (Seminar report after Monod-Ozawa, JFA 2010)