Transference for Banach space representations of nilpotent Lie groups. Part 1. Irreducible representations

TL;DR

This paper extends the CCR property from unitary to certain Banach space representations of nilpotent Lie groups, highlighting the role of reflexivity and employing transference methods and smooth vectors.

Contribution

It establishes a CCR property for irreducible representations on reflexive Banach spaces and shows failure on non-reflexive spaces, advancing understanding of representation theory.

Findings

CCR property holds for reflexive Banach space representations

Failure of CCR property in non-reflexive Banach spaces

Use of transference and smooth vectors in proofs

Abstract

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.