14 research outputs found
Heisenberg modules as function spaces
Let be a closed, cocompact subgroup of , where
is a second countable, locally compact abelian group. Using localization of
Hilbert -modules, we show that the Heisenberg module
over the twisted group -algebra
due to Rieffel can be continuously and densely embedded into the Hilbert space
. This allows us to characterize a finite set of generators for
as exactly the generators of multi-window
(continuous) Gabor frames over , a result which was previously known
only for a dense subspace of . We show that
as a function space satisfies two properties that
make it eligible for time-frequency analysis: Its elements satisfy the
fundamental identity of Gabor analysis if is a lattice, and their
associated frame operators corresponding to are bounded.Comment: 24 pages; several changes have been made to the presentation, while
the content remains essentially unchanged; to appear in Journal of Fourier
Analysis and Application
A dynamical approach to non-uniform density theorems for coherent systems
We introduce a notion of covolume for point sets in locally compact groups
that simultaneously generalizes the covolume of a lattice and the reciprocal of
the Beurling density for amenable, unimodular groups. This notion of covolume
arises naturally from the transverse measure theory of the associated hull
dynamical system of a point set. Using groupoid techniques, we prove a density
theorem for coherent frames over unimodular groups using this new notion of
covolume that generalizes both the previous theorems for uniform sampling in
general unimodular groups and those for non-uniform sampling in compactly
generated groups of polynomial growth. This density theorem also covers
important new examples, in particular model sets arising from cut-and-project
schemes.Comment: 30 page
Time-frequency analysis on the adeles over the rationals
We show that the construction of Gabor frames in with
generators in and with respect to time-frequency
shifts from a rectangular lattice is
equivalent to the construction of certain Gabor frames for over the
adeles over the rationals and the group .
Furthermore, we detail the connection between the construction of Gabor frames
on the adeles and on with the construction of
certain Heisenberg modules.Comment: minor revisions, added more references, added a Balian-Low type
result in the form of Proposition 4.
Smooth lattice orbits of nilpotent groups and strict comparison of projections
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 -module. An important ingredient in the
approach is that twisted group -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
-algebra has strict comparison of projections.Comment: 36 page
Deformations of Gabor frames on the adeles and other locally compact abelian groups
We generalize Feichtinger and Kaiblinger's theorem on linear deformations of
uniform Gabor frames to the setting of a locally compact abelian group .
More precisely, we show that Gabor frames over lattices in the time-frequency
plane of with windows in the Feichtinger algebra are stable under small
deformations of the lattice by an automorphism of . The
topology we use on the automorphisms is the Braconnier topology. We
characterize the groups in which the Balian-Low theorem for the Feichtinger
algebra holds as exactly the groups with noncompact identity component. This
generalizes a theorem of Kaniuth and Kutyniok on the zeros of the Zak transform
on locally compact abelian groups. We apply our results to a class of
number-theoretic groups, including the adele group associated to a global
field.Comment: 37 page
The density theorem for projective representations via twisted group von Neumann algebras
We consider converses to the density theorem for square-integrable, irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that the restriction of such a σ-projective unitary representation πof a unimodular, second-countable group G to a lattice Γ extends to a Hilbert module over the twisted group von Neumann algebra of (Γ, σ). We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner’s condition, we show that the center-valued von Neumann dimension reduces to the scalar value dπ vol(G/Γ), where dπ is the formal dimension of π and vol(G/Γ) is the covolume of Γ in G. We apply our results to characterize the existence of multiwindow super frames and Riesz sequences associated to π and Γ. In particular, we characterize when a lattice in the time-frequency plane of a second-countable, locally compact abelian group admits a Gabor frame or Gabor Riesz sequence
Heisenberg modules and Balian–Low theorems - Applications of operator algebras to Gabor analysis
The main way to study a periodic signal is to decompose it into a sum of simple signals, namely sine waves. However, when a signal changes substantially over time, such as a piece of music, different methods are needed. One method is to use Gabor frames. A Gabor frame represents a given signal in a way that emphasizes the signal’s frequency content at each point in time. For instance, a Gabor frame will represent an audio signal analogously to how sheet music is written.
Constructing good Gabor frames is not an easy task, and this problem has connections to many other areas in mathematics. In my dissertation, I have connected this problem to an area called operator algebras. A basic theorem about Gabor frames is the Balian-Low theorem, which is rooted in the uncertainty principle from quantum mechanics. I have shown that this theorem has a conceptual interpretation in operator algebras. Moreover, one can generally talk about Gabor frames in an abstract setting, namely in the context of an abelian topological group. I have completely classified the groups to which the Balian-Low theorem extends. One of the groups to which it extends is the rational adele group from number theory
Free actions of polynomial growth Lie groups and classifiable C*-algebras
We show that any free action of a connected Lie group of polynomial growth on a finite dimensional locally compact space has finite tube dimension. This is shown to imply that theassociated crossed product C*-algebra has finite nuclear dimension. As an application we showthat C*-algebras associated with certain aperiodic point sets in connected Lie groups of polynomial growth are classifiable. Examples include cut-and-project sets constructed from irreduciblelattices in products of connected nilpotent Lie groups.
On sufficient density conditions for lattice orbits of relative discrete series
This note provides new criteria on a unimodular group G and a discrete series representation (π, Hπ) of formal degree dπ> 0 under which any lattice Γ ≤ G with vol(G/Γ)dπ≤1 (resp. vol(G/Γ)dπ≥1) admits g∈ Hπ such that π(Γ) g is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.Analysi