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arXiv · 2107.13850

Smooth lattice orbits of nilpotent groups and strict comparison of projections

Erik Bédos·Ulrik Enstad·Jordy Timo van Velthoven
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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.

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Erik Bédos, Ulrik Enstad, Jordy Timo van Velthoven. 2021-07-29. Smooth lattice orbits of nilpotent groups and strict comparison of projections. https://doi.org/10.1016/j.jfa.2022.109572

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