Gheysens, MaximeMonod, Nicolas2017-05-012017-05-012017-05-01201710.24033/asens.2317https://infoscience.epfl.ch/handle/20.500.14299/136665WOS:000397183100004Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact a-compact groups (e.g., countable groups). Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau-Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for locally compact groups. We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.text::journal::journal article::research article