  We establish a general CCR (liminarity) property for uniformly bounded
irreducible representations of nilpotent Lie groups on reflexive Banach spaces,
extending the well known property of unitary irreducible representations of
these groups on Hilbert spaces. We also prove that this conclusion fails for
many representations on non-reflexive Banach spaces. Our approach to these
results blends the method of transference from abstract harmonic analysis and a
systematic use of spaces of smooth vectors with respect to Lie group
representations.
