Heisenberg modules as function spaces

Let $\Delta$ be a closed, cocompact subgroup of $G \times \widehat{G}$, where $G$ is a second countable, locally compact abelian group. Using localization of Hilbert $C^*$-modules, we show that the Heisenberg module $\mathcal{E}_{\Delta}(G)$ over the twisted group $C^*$-algebra $C^*(\Delta,c)$ due to Rieffel can be continuously and densely embedded into the Hilbert space $L^2(G)$. This allows us to characterize a finite set of generators for $\mathcal{E}_{\Delta}(G)$ as exactly the generators of multi-window (continuous) Gabor frames over $\Delta$, a result which was previously known only for a dense subspace of $\mathcal{E}_{\Delta}(G)$. We show that $\mathcal{E}_{\Delta}(G)$ 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 $\Delta$ is a lattice, and their associated frame operators corresponding to $\Delta$ 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

Gabor Duality Theory for Morita Equivalent C∗C^*-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule we introduce $(n,d)$-matrix frames, which generalize superframes and multi-window frames. Density theorems for $(n,d)$-matrix frames are established, which extend the ones for multi-window and super Gabor frames. Our approach is based on the localization of a Hilbert $C^*$-module with respect to a trace.Comment: 36 page

Detecting ideals in reduced crossed product C*-algebras of topological dynamical systems

We introduce the $\ell^1$-ideal intersection property for crossed product C*-algebras. It is implied by C*-simplicity as well as C*-uniqueness. We show that topological dynamical systems of arbitrary lattices in connected Lie groups, arbitrary linear groups over the integers in a number field and arbitrary virtually polycyclic groups have the $\ell^1$-ideal intersection property. On the way, we extend previous results on C*-uniqueness of $\mathrm{L}^1$-groupoid algebras to the general twisted setting.Comment: 23 pages; v2 added full proof for Lemma 6.1, minor correction

Polynomial growth and property RDpRD_p for \'etale groupoids with applications to KK-theory

We investigate property $RD_p$ for \'etale groupoids and apply it to $K$-theory of reduced groupoid $L^p$-operator algebras. In particular, under the assumption of polynomial growth, we show that the $K$-theory groups for a reduced groupoid $L^p$-operator algebra is independent of $p\in (1, \infty)$. We apply the results to coarse groupoids and graph groupoids.Comment: Comments welcom

The Algebraic Bivariant Connes-Chern Character

In this thesis we present many properties of bivariant periodic cyclic homology with the purpose of then constructing two bivariant Connes-Chern characters from algebraic versions of Kasparov's KK-theory with values in bivariant periodic cyclic homology. The thesis is naturally divided into three parts. In the first part, which spans the two first chapters, periodic cyclic theory is presented, starting with the very basic definitions in cyclic theory. The properties of differential homotopy invariance, Morita invariance, and excision, all of which are important for the construction of bivariant Connes-Chern characters, are discussed. In the second part we discuss algebraic KK-theory based on the reformulations of Kasparov's KK-theory by Cuntz and Zekri. By using the properties of bivariant periodic cyclic theory from the first part, we construct two different bivariant Connes-Chern characters. In the third part we discuss possible extensions of the theory to topological algebras, in particular a well-behaved class of topological algebras known as m-algebras

Twisted Convolution Algebras and Applications to Gabor Analysis

This thesis concerns several aspects of twisted convolution algebras, with a particular focus on problems arising in Gabor analysis. A significant portion of the thesis is dedicated to the study of Hilbert C -modules known as Heisenberg modules and how they relate to Gabor frame theory. This relation showcases the link between finite Hilbert C -module frames and Gabor frames. Further, the thesis concerns certain properties of twisted convolution algebras of locally compact groups, in particular spectral invariance and C -uniqueness, and we find use for both these properties in Gabor analysis. The problem of C -uniqueness is also considered for the case of twisted convolution algebras of second-countable locally compact Hausdorff étale groupoids

Modulation spaces as a smooth structure in noncommutative geometry

We demonstrate how to construct spectral triples for twisted group C^*-algebras of lattices in phase space of a second-countable locally compact abelian group using a class of weights appearing in time–frequency analysis. This yields a way of constructing quantum C^k-structures on Heisenberg modules, and we show how to obtain such structures using Gabor analysis and certain weighted analogues of Feichtinger’s algebra. We treat the standard spectral triple for noncommutative 2-tori as a special case, and as another example we define a spectral triple on noncommutative solenoids and a quantum C^k-structure on the associated Heisenberg modules

C∗-uniqueness Results for Groupoids

For a 2nd-countable locally compact Hausdorff étale groupoid G with a continuous 2-cocycle σ we find conditions that guarantee that ℓ1(G,σ) has a unique C∗-norm