Smooth lattice orbits of nilpotent groups and 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 $C^{\ast }$-module. An important ingredient in the approach is that twisted group $C^{\ast }$-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 $C^{\ast }$-algebra has strict comparison of projections.

36 pages

References in corpus (22)

• Frames in Hilbert C*-modules and C*-algebras
• Quasidiagonality of nuclear C*-algebras
• The stable and the real rank of Z-absorbing C*-algebras
• Gabor Frames and Totally Positive Functions
• Sampling Theorems for Shift-invariant Spaces, Gabor Frames, and Totally Positive Functions
• Deformation of Gabor systems
• Modular curvature and Morita equivalence
• Quasitraces on exact C*-algebras are traces
• Density of Sampling and Interpolation in Reproducing Kernel Hilbert Spaces
• The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2 (Z)
• K-Theory for operator algebras. Classification of C${}^{\ast }$-algebras
• Phase Space Localization of Riesz bases for L^2(R^d)
• Strict comparison and Z-absorption of nuclear C*-algebras
• Primeness and primitivity conditions for twisted group $C^{\ast }$-algebras
• Heisenberg modules as function spaces
• The density theorem for projective representations via twisted group von Neumann algebras
• $C^{\ast }$-algebras generated by projective representations of free nilpotent groups
• Density results for continuous frames
• New Function Spaces Associated to Representations of Nilpotent Lie Groups and Generalized Time-Frequency Analysis
• Discrete Cocompact subgroups of G_53 and related C*-algebras
• Twisted Fourier(-Stieltjes) spaces and amenability
• On Projective Representations of Finitely Generated Groups