Abstracts

The main theme of the report is a combinatorial description of derivations on a group algebra. The linear mapping $d:{\bf C}[G]\to{\bf C}[G]$ is said to be the derivation if the Leibniz rule hold $$d(uv) = d(u)v+u d(v), \forall u,v\in{\bf C}[G].$$ We can consider a gruppoid $\Gamma$ in the following way.
 
The set of objects is a set of elements $Obj(\Gamma) = \{g\in G\}$. The set of arrows is a set of pairs $Hom(\Gamma):=\{(u, v)|u,v\in G\}$. An arrow $\phi = (u,v)$ has a source $s(\phi)= v^{-1}u$ and a target $t(\phi) = uv^{-1}$. For arrows $\phi = (u_1,v_1)$ and $\psi = (u_2, v_2)$ such that $t(\phi)=s(\psi)$ we will define a composition $$\psi\circ\phi := (v_2 u_1,v_2 v_1).$$

We will call a character any mapping $\chi:Hom(\Gamma)\to {\bf C}$ such that $\chi(\psi\circ\phi)=\chi(\psi)+\chi(\phi)$ when $t(\phi)=s(\psi)$. We will call a character a locally finite character if $\forall v\in G \chi ((u,v)) = 0$, for all except of finite number of elements $u\in G$.

Theorem. For any derivation $d$ exist a character $\chi_d$ such that $\forall a\in G$ $$d(a) = a\left(\sum\limits_{t\in [u]} \chi_d(at, a)t  \right).$$

We get a new look at the study of differentiations from the position of categories and their characters. It becomes reasonable to define the concept of quasiinner derivations $QInn$, as a derivations which characters are trivial on loops. It turns out that such differentiations form an ideal. This gives an interesting connection with one old topological invariant, namely with the ends of topological spaces.  And let $\Gamma$ be the conjugacy diagram for a given presentation of a group. Understanding this graph as a metric space we denote by $e(\Gamma)$ the number of ends of this space.

Theorem. The number $e(\Gamma)$ doesn't depend on a corepresentation of the group $G$. And the following formula holds $$\dim(QInn/Inn) = e(\Gamma)-1.$$

This approach, among other things, allows us to calculate quasiouter derivations $QOut:=Der/QInn$. It turns out that such differentiations can also be understood as characters on a $2$-groupoid.

The pillars of Gabor analysis are the Janssen representation, the duality principle and the Wexler-Raz biorthogonality relations, which together make up the duality theory of Gabor analysis. We present a vast generalization of duality theory to a class of Hilbert $C^*$-modules that we refer to as Gabor bimodules. Our methods rely on the Cohen-Hewitt factorization theorem for Banach modules and the Morita equivalence of $C^*$-algebras, which are implemented by equivalence bimodules. In short, Gabor bimodules are equivalence bimodules with some additional features.

We show that our abstract duality theory reduces to the well-known facts about Gabor frames for locally compact abelian groups if one uses some well-known facts about twisted group algebras and Hilbert $C^*$-modules.

Furthermore, we discuss a generalization of the notion of $n$-multiwindow and $d$-super Gabor frames, namely $(n,d)$-matrix Gabor frames and their relation to projections in matrix algebras over $C^*$-algebras.

Parts of this work are joint work with Mads S. Jakobsen and Franz Luef.

The operator representation of frames is a new subject of frame theory related to dynamical sampling. Given a frame $\{f_n\}$ of a Hilbert space the aim is to find a bounded operator whose orbit on some element is exactly $\{f_n\}$. The iteration of the operator can be over the natural numbers or over all the integer numbers. The topic is strictly connected to compressed shifts in model spaces.

Classical frames of $L^2(\mathbb{R})$ are Gabor frames (with one window) that have the form $$\mathcal{G}(g,a,b)=\{e^{2\pi i m b x}g(x-na)\}_{n,m\in \mathbb{Z}},$$ for some $g\in L^2(\mathbb{R})$ and $a,b\gt 0$. 

We will focus on representation with orbits over integer numbers and on the following question: are there Gabor frames which are orbits of a bounded bijective operator?

The Balian-Low Theorem is a classical result in Gabor analysis on the non-existence of Gabor frames with sufficiently well-localized windows at the critical density. However, the Balian-Low Theorem concerns critical sampling of Gabor atoms on the real line $\mathbb{R}$, and Gröchenig showed that there is no analogue of the theorem for general locally compact abelian groups. One can then ask the question: For which groups does one obtain an analogue of the Balian-Low Theorem?

In this talk, I will show that the validity of a Balian-Low type statement in a given second countable, locally compact abelian group $G$ is equivalent to the nontriviality of an associated vector bundle. When $G = \mathbb{R}$, we recover the usual Balian-Low Theorem for the Feichtinger algebra, and the associated vector bundle has as base space the $2$-torus and is closely related to a bundle due to Balan. We also obtain a Balian-Low type Theorem in the setting of the group $G = \mathbb{R} \times \mathbb{Q}_p$, where $\mathbb{Q}_p$ denotes the $p$-adic numbers, by analyzing the associated vector bundle.

The proof uses the Heisenberg modules due to Rieffel, which have connections to Gabor analysis as shown by Jakobsen and Luef. Specifically, the proof implements the Zak transform for locally compact abelian groups as an isomorphism of Hilbert $C^*$-modules.

In this talk I would like to discuss the problem of approximating continuous problems in time-frequency and Gabor analysis by means of finite dimensional computations using mathematical software, such as MATLAB™. Unlike engineers, who switch easily from a heuristic argument concerning functions on $\mathbb{R}$ or $\mathbb{R}^d$ involving the Fourier transform easily to the use of the FFT (Fast Fourier Transform), using simply the argument that “the computer allows only to work with finite length vectors”, we see it as a modelling and conceptual problem.

In the paper [3] and in several talks we have presented the idea of Conceptual Harmonic Analysis, which has the goal to conceal the ideas from Abstract Harmonic Analysis (AHA) with ideas from Numerical or Computational Harmonic Analysis (NCHA). AHA by itself allows to understand the analogies between different concrete locally compact Abelian (LCA) groups $G$. There is always an invariant Haar measure, a dual group $\widehat{G}$, which consists of all the characters of $G$, the so-called pure frequencies or plane waves. This opens the way to do time-frequency analysis, using the combination of translation and modulation operators and the STFT (Short Time Fourier Transform). The time-frequency approach also allows to introduce so-called mild distributions, which form the dual of the Segal algebra $S_0(G)$. Over $\mathbb{R}^d$ one can characterize $S_0^\prime(G)$ as the space of all tempered distributions which have a bounded STFT, and $w^*$-convergence of sequences in $S_0^\prime(G)$ corresponds to uniform convergence of the STFT over compact subsets (or equivalently pointwise convergence).

In contrast, NCHA tries to find efficient algorithms in order to realize a given task. The most famous routine is of course the FFT, an efficient implementation of the DFT (discrete Fourier transform). In discrete Gabor analysis we have many fast algorithms, to find the dual Gabor atom for a given Gabor frame, or to compute the best approximation of a give matrix (in the Frobenius norm). In each of these settings the group is fixed, i.e. on compares the speed or computational efficiency for a given signal length, maybe checking in addition how the computational costs increase with the signal length. For the FFT one has the famous order $ n \log(n)$, to give an example.

The talk will address the problem of approximating a continuous problem by a sequence (or in fact net) of finite dimensional problems of the same kind. The key idea is that of structure preserving approximation. The key questions are of the following kind:

• Given a continuous problem over $\mathbb{R}$, what is a good choice of a corresponding finite model that helps to approximate (in a suitable sense) the problem in the best way?
• What are the computational costs for a good (approximate) solution in the finite-dimensional setting, and how can one relate the continuous and discrete problem appropriately?

Unlike classical approximation theory (and even constructive approximation theory) we have, at least a priori, not even the possibility to measure the distance between the finite approximation and the continuous limit, because they appear to “live” on different groups. Still there are some very natural ideas, to be explained in the talk.

For comparison, let us think about Riemann integrals $\int_a^b f(x)dx$. The integral is the “continuous limit” which exists for any continuous function $f$ on $[a,b]$, thanks to the uniform continuity of $f$ on a compact interval. This is OK, but in practice one may prefer to choose an efficient numerical integration method in order to quickly obtain a very good approximation to the required numerical value, but only for functions satisfying certain smoothness requirements.

Concretely, we propose to approximate functions on $\mathbb{R}$ (for simplicity) by finite sampling sequences of length $n = k^2$, obtained from a $k$-periodic version of $f$, sampled at the rate $1/k$. In this way, for example, any Fourier-invariant function will be mapped on a finite sequence which is (up to the constant $\sqrt{n}$) invariant under the FFT. For $f \in S_0(\mathbb{R})$ an approximate inverse can be obtained by piecewise linear interpolation. This approach provides a natural embedding of signals of length $n$ into those of length $4n$ (or $9n$, etc.), similar to Riemannian sums obtain by continued bi-section.

We will illustrate concrete examples via the computation of the Fourier transform of $f \in S_0(\mathbb{R})$ via FFT algorithms, or the determination of dual Gabor atoms via finite Gabor code. More recently we are interested in different methods to compute discrete Hermite functions. This example allows us to discuss the problem: What is a good discrete Hermite function?

References

1. H. G. Feichtinger and N. Kaiblinger. Quasi-interpolation in the Fourier algebra. J. Approx. Theory, 144(1):103-118, 2007.
2. N. Kaiblinger. Approximation of the Fourier transform and the dual Gabor window. J. Fourier Anal. Appl., 11(1):25-42, 2005.
3. H. G. Feichtinger. Thoughts on Numerical and Conceptual Harmonic Analysis. New Trends in Applied Harmonic Analysis. pg. 301-329. Birkhäuser, 2016.

There is an abstract version of the Gabor systems duality principle for  group representations, and it is known that this duality principle  has some connections with the classification problem for free-group von Neumann algebras. In this talk I will revisit this general duality principle, and discuss some recent both trivial and nontrivial observations that lead to a generalization of the Wexler-Raz biorthogonality and the Fundamental Identity for Gabor representations to general group representations. Additionally,  I will also discuss a duality principle connecting the  “super-frames” and “multi-frames” through a commutant dual pair of group representations, and discuss its applications to time-frequency analysis. 

We show that the operator algebra products a given operator space can be equipped with are precisely the bilinear mappings on the operator space that are implemented by contractive quasi-multipliers. We also characterize the operator algebra products in terms of only matrix norms and completely contractive mappings using the Haagerup tensor product. That is, we show that a geometric property (matrix norm) induces an algebraic property (product).

As application, we study extreme points of the unit ball of an operator space by introducing the new notion (approximate) quasi-identities. More specifically, we characterize an operator algebra having a contractive (approximate) quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points of the unit ball (of the weak*-closure) of the underlying operator space. Furthermore, we give a necessary and sufficient condition for a given operator space to be qualified to become a $C^\ast$-algebra or a one-sided ideal in a $C^\ast$-algebra in terms of quasi-multipliers.

The quantum Heisenberg manifolds (QHMs) are noncommutative manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds. The QHMs give another important family of noncommutative manifolds along with noncommutative two-tori. In this talk, we first review noncommutative Yang-Mills theory on the QHMs based on the framework given by Connes and Rieffel and then we discuss recent results of Yang-Mills connections with constant curvature on the QHMs. This is a joint work with Franz Luef and Judith Packer.

A unital $C^*$-algebra is a noncommutative generalization of the algebra $C(X)$ of continuous functions on a compact topological space; in the nonunital case, it is a generalization of the algebra $C_0 (X)$ of continuous functions that vanish at infinity, or more accurately, functions that are each arbitrarily small outside a sufficient large compact set. Let us consider what might be the proper generalization of compactly supported functions to the noncommutative case. The usual approach in noncommutative geometry is to fix some holomorphically closed subalgebra and to use that, but what happens if we need more information about the "compactly supported functions"? The answer has possibly been found by Pedersen, in the form of the Pedersen ideal. However, there are some subtle questions that arise when we take tensor products. One would expect the Pedersen ideal of the tensor product of two $C^*$-algebras to “multiply” in an obvious sense. However, while this might be expected, the situation is complicated by the fact that there is, in general, more than one $C^*$-norm that one can put on the tensor product of $C^*$-algebras. We show that positive elements of a Pedersen ideal of a tensor product can be approximated in a particularly strong sense by sums of tensor products of positive elements. We show that the positive elements of a Pedersen ideal are sometimes stable under Cuntz equivalence. We generalize a result of Pedersen's by showing that certain classes of completely positive maps take a Pedersen ideal into a Pedersen ideal. This has a range of applications to the structure of tracial cones and related topics, such as the Cuntz-Pedersen space or the Cuntz semigroup. For example, we determine the cone of lower semi-continuous traces of a tensor product in terms of the traces of the tensor factors, in an arbitrary $C^*$-tensor norm.

In this talk, I present the basic ideas of Gabor analysis and its relation to Heisenberg modules over noncommutative tori. There is a dictionary between basic notions in Gabor analysis and noncommutative geometry which allows one to reformulate problems in the respective areas in a novel way. I discuss some entries of this dictionary.  

We generalize three main concepts of Gabor analysis for lattices to the setting of model sets: Fundamental Identity of Gabor Analysis, Janssen’s representation of the frame operator and Wexler-Raz biorthogonality relations utilizing the connection between model sets and almost periodic functions, as well as Poisson’s summation formula for model sets.

In the high-energy quantum-physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. More recently I have dealt with corresponding statements in the literature about vector bundles on spheres and matrix algebras. I will very briefly indicate how some of this works.

But physicists want, even more, to treat structures on spheres (and other spaces) such as Dirac operators, Yang-Mills functionals, etc., and they want to approximate these by corresponding structures on matrix algebras. In preparation for understanding what the Dirac operators should be have been working out what the corresponding "cotangent bundles" should be for the matrix algebras. (In the physics literature there are at least 3 inequivalent suggestions for the Dirac operators.) This is work in progress. I will report on what I have found by the time of the meeting. Because coherent states lie behind much of this, it might at some point have some relationship to Gabor Analysis.

Kernel theorems assert that every "reasonable" operator can written as a "generalized" integral operator. For instance, the Schwartz kernel theorem states that a continuous linear operator $A$ mapping Schwartz functions into tempered distributions possesses a unique distributional kernel $K\in \mathcal{S}^\prime(\mathbb{R}^{2d})$, such that $ \langle Af,g\rangle= \langle K,g\otimes f \rangle,\  f,g\in\mathcal{S}(\mathbb{R}^d).$  During this talk we will present a similar kernel theorem for operators acting between coorbit spaces of test functions and their distribution spaces. Loosely speaking, a coorbit space is a Banach space of distributions described by the behavior of its generalized wavelet transform (generated by an integrable group representation). With this approach, we are able to recover known results such as Feichtinger's kernel theorem and, using a version of Schur's test, to characterize boundedness between large classes of coorbit spaces. This is not a mere abstraction and generalization as the specific choice of the group representation  implies explicit kernel theorems for well-known spaces such as Besov spaces or modulation spaces.