Smooth lattice orbits of nilpotent groups and strict comparison of projections

TL;DR

This paper establishes density conditions for smooth vectors forming frames in lattice orbits of nilpotent group representations, linking harmonic analysis with operator algebra properties like strict comparison of projections.

Contribution

It introduces new density criteria for smooth frames in nilpotent group lattice orbits and connects these to the finite nuclear dimension of associated twisted group C*-algebras.

Findings

Density conditions for smooth vectors in lattice orbits

Balian-Low type theorems at critical density

Finite nuclear dimension implies strict comparison of projections

Abstract

This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions involve the product of lattice co-volume and formal dimension, and complement Balian-Low type theorems for the non-existence of smooth frames and Riesz sequences at the critical density. The proof hinges on a connection between smooth lattice orbits and generators for an explicitly constructed finitely generated Hilbert $C^*$-module. An important ingredient in the approach is that twisted group $C^*$-algebras associated to finitely generated nilpotent groups have finite decomposition rank, hence finite nuclear dimension, which allows us to deduce that any matrix algebra over such a simple $C^*$-algebra has strict comparison of projections.