Smooth lattice orbits of nilpotent groups and strict comparison of projections

Erik Bédos*, Ulrik Enstad, Jordy Timo van Velthoven

*Corresponding author for this work

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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.

Original languageEnglish
Article number109572
Number of pages48
JournalJournal of Functional Analysis
Volume283
Issue number6
DOIs
Publication statusPublished - 2022

Keywords

  • Decomposition rank
  • Frame
  • Projective module
  • Smooth vector

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