Paweł Kasprzak

Rieffel deformation via crossed products

We start from Rieffel data (A,Ψ,ρ), where A is a C∗-algebra, ρ is an action of an abelian group Γ on A and Ψ is a 2-cocycle on the dual group. Using Landstad theory of crossed product we get a deformed C∗-algebra AΨ. In the case of Γ=Rn we obtain a very simple proof of invariance of K-groups under the deformation. In the general case we also get a very simple proof that nuclearity is preserved under the deformation. We show how our approach leads to quantum groups and investigate their duality. The general theory is illustrated by an example of the deformation of SL(2,C). A description of it, in terms of noncommutative coordinates αˆ,βˆ,γˆ,δˆ, is given.

EMBEDDABLE QUANTUM HOMOGENEOUS SPACES

Abstract We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we recall an interesting duality for such objects studied earlier by M. Izumi, R. Longo, S. Popa for compact Kac algebras and by M. Enock in the general case of locally compact quantum groups. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles. The former turns out to be an interesting application of the duality mentioned above.

Pitfalls in compressed sensing reconstruction and how to avoid them

Multidimensional NMR can provide unmatched spectral resolution, which is crucial when dealing with samples of biological macromolecules. The resolution, however, comes at the high price of long experimental time. Non-uniform sampling (NUS) of the evolution time domain allows to suppress this limitation by sampling only a small fraction of the data, but requires sophisticated algorithms to reconstruct omitted data points. A significant group of such algorithms known as compressed sensing (CS) is based on the assumption of sparsity of a reconstructed spectrum. Several papers on the application of CS in multidimensional NMR have been published in the last years, and the developed methods have been implemented in most spectral processing software. However, the publications rarely show the cases when NUS reconstruction does not work perfectly or explain how to solve the problem. On the other hand, every-day users of NUS develop their rules-of-thumb, which help to set up the processing in an optimal way, but often without a deeper insight. In this paper, we discuss several sources of problems faced in CS reconstructions: low sampling level, missassumption of spectral sparsity, wrong stopping criterion and attempts to extrapolate the signal too much. As an appendix, we provide MATLAB codes of several CS algorithms used in NMR. We hope that this work will explain the mechanism of NUS reconstructions and help readers to set up acquisition and processing parameters. Also, we believe that it might be helpful for algorithm developers.

Rieffel Deformation of Group Coactions

Let G be a locally compact group, Γ ⊂ G an abelian subgroup and let Ψ be a continuous 2-cocycle on the dual group Γ̂. Let B be a C-algebra and ∆B ∈ Mor(B,B⊗C0(G)) a continuous right coaction. Using Rieffel deformation, we can construct a quantum group (C0(G),∆) and the deformed C -algebra B. The aim of this paper is to present a construction of the continuous coaction ∆ B of the quantum group (C0(G),∆) on B. The transition from the coaction ∆B to its deformed counterpart ∆ Ψ B is nontrivial in the sense that ∆ B contains complete information about ∆B. In order to illustrate our construction we apply it to the action of the Lorentz group on the Minkowski space obtaining a C-algebraic quantum Minkowski space.

Tracking algorithm in the multiband PCL-PET fusion system

In this article we discuss certain aspects of the algorithm for data processing in the PCL - PET passive radar system. We discuss the part of the fusion algorithm which is used for object tracking and estimate improvement. We also propose a method for bistatic measurements classification using the predicted object position and covariance matrices. Proposed algorithm is also suitable for asynchronous and random flow of input measurements. Presented algorithms, which are a work-inprogress, are being implemented and will be subject to further experiments and development on a PCL-PET system prototype.

On a Certain Approach to Quantum Homogeneous Spaces

We propose a definition of a quantum homogeneous space of a locally compact quantum group. We show that classically it reduces to the notion of homogeneous spaces, giving rise to an operator algebraic characterization of the transitive group actions. On the quantum level our definition goes beyond the quotient case providing a framework which, besides the Vaes’ quotient of a locally compact quantum group by its closed quantum subgroup (our main motivation) is also compatible with, generically non-quotient, quantum homogeneous spaces of a compact quantum group studied by P. Podleś as well as the Rieffel deformation of G-homogeneous spaces. Finally, our definition rules out the paradoxical examples of the non-compact quantum homogeneous spaces of a compact quantum group.

The Heisenberg–Lorentz quantum group

In this article we present a new deformation of SL(2, C) on the C � -algebra level. The method of its construction is the Rieffel deformation. We give a detailed description of so obtained a quantum group (A,�) in terms of affiliated elements ˆ �, ˆ �, ˆ , ˆ 2 Agenerating A. In order to construct ˆ �, we split A onto parts on which ˆ � can easily be defined and then show that gluing procedure can be performed. Using the duality for locally compact quantum groups we were able to describe all representations of the C � -algebra A and to analyze the action of � on generators ˆ ˆ �, ˆ ˆ �.

QUANTUM AUTOMORPHISM GROUPS OF FINITE QUANTUM GROUPS ARE CLASSICAL

Abstract In a recent paper of Bhowmick, Skalski and Soltan the notion of a quantum group of automorphisms of a finite quantum group was introduced and, for a given finite quantum group G , existence of the universal quantum group acting on G by automorphisms was proved. We show that this universal quantum group is in fact a classical group. The key ingredient of the proof is the use of multiplicative unitary operators, and we include a thorough discussion of this notion in the context of finite quantum groups.

Position estimation in the multiband PCL-PET fusion system

In this article we discuss certain aspects of the algorithm for data processing in the PCL - PET passive radar system.We discuss the initial part of the fusion algorithm the main function of which is to provide the number of objects, their position estimations and the covariance matrix of these estimates. The measurement data model and processing scheme is presented and discussed. Presented fusion algorithms for dislocated sensors have been implemented and tested. Obtained results illustrate performance of overcoming difficulties arising in typical multi-static systems - ghost objects elimination and accurate position estimation. Presented algorithms are a work in progress and are being implemented and will be subject to further experiments and development on a PCL-PET system prototype.

Objects recognition with high-Resolution InSAR data and Global Geometric Feature Map

Modern airborne or satellite SAR radar systems provide geometric resolution below ten centimeters. By SAR interferometry from pairs of such images, DEM image can be obtained. Using data of this kind it is possible to build a simple 3D models of remote objects. Similarity measurement between model of unknown object and known database models can be used for classification and recognition. In this paper Global Geometric Feature Map method with improvements is discussed. Since 3D polygonal model can be expressed as a set of facets, the GGFM can fast constitute a spherical transformation to new feature vector containing: normal orientation, area and position of every facet on the model surface. Offline analysis can be done by means of computation of spherical correlation between the GGFM of the object and the models. For online analysis (e.g.: automatic fire-control systems) simplifications of correlation algorithm are required. It can be done by passing the a certain bitmap representation of the data. The experimental results are based on the MSTAR radar images database. In this article we provide the examples of the original GGFM and 2D GGFM analysis.