Uniformly Bounded Representation

A uniformly bounded representation of a discrete group on a complex Hilbert space is a group homomorphism into the bounded invertible operators satisfying

Here the norm is the operator norm. The common bound applies to every group element, rather than only a generating set. Every unitary representation on a nonzero Hilbert space has bound 1. A unitarizable group is one for which every such representation becomes unitary after one bounded invertible change of coordinates.