Final Report Summary - MOTIF (Modern Methods of Operator Algebras for Time-Frequency Analysis)
Project context and objectives
The connection between Gabor analysis and noncommutative geometry over noncommutative tori is the starting point for the research in this project. It allows us to transfer methods and results from one field to the other to enrich both fields. One of the main goals of this project is to gain a better understanding of the structures in time-frequency analysis, and especially in Gabor analysis, by invoking results on projective modules over operator algebras, specifically the K-theory of operator algebras and Morita-Rieffel equivalence of operator algebras. Furthermore we want to investigate the relation between time-frequency analysis and quantum mechanics from the perspective of operator algebras; in particular our focus is towards deformation quantisation and phase-space quantum mechanics.
The project is progressing as outlined in the proposal. According to the interdisciplinary nature of the project, I have conducted research on the frontiers of noncommutative geometry, quantum mechanics and time-frequency analysis. The main objective is the use of modern methods of operator algebras to gain a deeper understanding of time-frequency analysis and quantum mechanics. The arsenal of tools I am invoking in this work is coming from noncommutative geometry, the theory of Hilbert C*-modules and K-theory. In particular I was able to translate and generalise the construction of projective modules over noncommutative tori, which is one of the highlights of noncommutative geometry, into the framework of Gabor analysis. Therefore, I have now produced a dictionary that relates notions from noncommutative geometry with ones in time-frequency analysis. Consequently, the very abstract theory of noncommutative geometry is connected in a novel way with real-world concepts and problems.
In the paper 'Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces', I displayed a hitherto unknown connection between time-frequency analysis and noncommutative geometry. This paper is one of the cornerstones of the present project and it has led to forthcoming publications like the work on quantum theta functions and the constructions of projections in noncommutative tori, where I was able to contribute to a further understanding of notions and objects in noncommutative geometry and operator algebras. The joint paper on quantum theta functions with Yuri I. Manin from the Max Planck Institute for Mathematics in Bonn was an exceptional opportunity and one of the highlights of the project. Since it was an extraordinary experience to work with one of the greatest mathematicians of the 20th century.
Further work addressed topics in quantum mechanics. In joint work with Maurice de Gosson, we pointed out the relevance of methods from time-frequency analysis for deformation quantisation, especially in the class of modulation spaces.
Project results
The connection between Gabor analysis and noncommutative geometry over noncommutative tori is the starting point for the research in this project. It allows us to transfer methods and results from one field to the other to enrich both fields. One of the main goals of this project is to gain a better understanding of the structures in time-frequency analysis, and especially in Gabor analysis, by invoking results on projective modules over operator algebras, specifically the K-theory of operator algebras and Morita-Rieffel equivalence of operator algebras. Furthermore, we want to investigate the relation between time-frequency analysis and quantum mechanics from the perspective of operator algebras; in particular our focus is towards deformation quantisation and phase-space quantum mechanics.
More concretely, we have been able to provide a dictionary between Gabor analysis and noncommutative geometry over noncommutative tori. The most important entries of this dictionary are: (a) Gabor frame operators are Hilbert C*-module operators for noncommutative tori; and (b) multi-window Gabor frames are projective modules over noncommutative tori, i.e. non-commutative vector bundles over noncommutative tori. These insights have turned out to be crucial for progress on the proposed objectives. On the one hand, they made it possible to prove the existence of multi-window Gabor frames with good quality (e.g. Schwartz functions), but on the other hand, they provided a way to extend the results of Connes and Rieffel from smooth noncommutative tori to the class of twisted group algebras via the use of methods from Gabor analysis.
Furthermore, we were able to develop a new approach to the K-theory of noncommutative tori and rotation algebras, since projections in noncommutative tori turned out to be generated by tight Gabor atoms. As a first application, I was able to construct new projections in noncommutative tori. Furthermore, these investigations demonstrate that the duality theory of Gabor frames has a natural description in the setting of Morita-Rieffel equivalence of non-commutative tori, in particular that the Wexler-Raz biorthogonality conditions are intrinsically related to projections in non-commutative tori. These investigations also shed new light on the quantum theta functions of Yuri I. Manin, and the well-known projections of Boca. In a two-dimensional setting, non-commutative tori may be viewed as rotation algebras. If one follows the dictionary, one arrives at an operator-algebraic approach to the Walnut representation of Gabor frames, and the conditions on the existence of projections in rotation algebras are related to well-known characterisation of Gabor frames in terms of the Walnut representation. Furthermore, this research demonstrates that the famous Rieffel projections in irrational rotation algebras are actually given by a special class of Gabor frames known as painless non-orthogonal Gabor expansions. Other main results show that a particular Wiener amalgam space provides a natural class of functions leading to projections in rotation algebras and an important result of Kristal and Okoudjou's work on Gabor frames implies the existence of well-localised projections in rotation algebras.
These results provided a way to clarify the relation between Rieffel's work on coupling constants among von Neumann algebras, and the density theorem of Gabor analysis: a proof of the Stone-von Neumann theorem based on methods from time-frequency analysis. The key of the argument is a description of projections in the twisted group algebra of the phase space, which might be considered as in the case of noncommutative tori, where the density of lattices is arbitrarily large. According to Rieffel's findings, this yields results on the structure of noncommutative tori over the adjoint noncommutative tori, which in this case are objects generated by time-frequency shifts over a lattice with an arbitrarily small density.
In addition, these investigations led to the question of when noncommutative tori are Morita-Rieffel equivalent to the continuous functions over the torus, since this is of predominant relevance for the application of Gabor frames to real-world problems. In a short note I was able to show that this is exactly the case if the lattice generating a Gabor frame equals its adjoint, and this is precisely the case if the lattice is symplectic.
Another step towards this objective is to work out the finite-dimensional case, which amounts to the construction of projections in matrix algebras. At the moment I am working out the structures in this setting and it turns out that these are closely connected to algorithms used in wireless communication. Consequently, the Morita-Rieffel equivalence of matrix algebras provides the theoretical framework for many algorithms designed by engineers in signal analysis, e.g. in the transmission of signals via cellular phones.
Finally, we worked on some aspects of noncommutative quantum mechanics in collaboration with two top experts of this theory, Dias and Prata, and with the help of Maurice de Gosson we were able to give a mathematically rigorous formulation. In these works we study noncommutative quantum mechanics from the point of view of deformation quantisation. These investigations have a natural connection to noncommutative geometry and the structure of noncommutative tori, which we want to address in subsequent work.
The connection between Gabor analysis and noncommutative geometry over noncommutative tori is the starting point for the research in this project. It allows us to transfer methods and results from one field to the other to enrich both fields. One of the main goals of this project is to gain a better understanding of the structures in time-frequency analysis, and especially in Gabor analysis, by invoking results on projective modules over operator algebras, specifically the K-theory of operator algebras and Morita-Rieffel equivalence of operator algebras. Furthermore we want to investigate the relation between time-frequency analysis and quantum mechanics from the perspective of operator algebras; in particular our focus is towards deformation quantisation and phase-space quantum mechanics.
The project is progressing as outlined in the proposal. According to the interdisciplinary nature of the project, I have conducted research on the frontiers of noncommutative geometry, quantum mechanics and time-frequency analysis. The main objective is the use of modern methods of operator algebras to gain a deeper understanding of time-frequency analysis and quantum mechanics. The arsenal of tools I am invoking in this work is coming from noncommutative geometry, the theory of Hilbert C*-modules and K-theory. In particular I was able to translate and generalise the construction of projective modules over noncommutative tori, which is one of the highlights of noncommutative geometry, into the framework of Gabor analysis. Therefore, I have now produced a dictionary that relates notions from noncommutative geometry with ones in time-frequency analysis. Consequently, the very abstract theory of noncommutative geometry is connected in a novel way with real-world concepts and problems.
In the paper 'Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces', I displayed a hitherto unknown connection between time-frequency analysis and noncommutative geometry. This paper is one of the cornerstones of the present project and it has led to forthcoming publications like the work on quantum theta functions and the constructions of projections in noncommutative tori, where I was able to contribute to a further understanding of notions and objects in noncommutative geometry and operator algebras. The joint paper on quantum theta functions with Yuri I. Manin from the Max Planck Institute for Mathematics in Bonn was an exceptional opportunity and one of the highlights of the project. Since it was an extraordinary experience to work with one of the greatest mathematicians of the 20th century.
Further work addressed topics in quantum mechanics. In joint work with Maurice de Gosson, we pointed out the relevance of methods from time-frequency analysis for deformation quantisation, especially in the class of modulation spaces.
Project results
The connection between Gabor analysis and noncommutative geometry over noncommutative tori is the starting point for the research in this project. It allows us to transfer methods and results from one field to the other to enrich both fields. One of the main goals of this project is to gain a better understanding of the structures in time-frequency analysis, and especially in Gabor analysis, by invoking results on projective modules over operator algebras, specifically the K-theory of operator algebras and Morita-Rieffel equivalence of operator algebras. Furthermore, we want to investigate the relation between time-frequency analysis and quantum mechanics from the perspective of operator algebras; in particular our focus is towards deformation quantisation and phase-space quantum mechanics.
More concretely, we have been able to provide a dictionary between Gabor analysis and noncommutative geometry over noncommutative tori. The most important entries of this dictionary are: (a) Gabor frame operators are Hilbert C*-module operators for noncommutative tori; and (b) multi-window Gabor frames are projective modules over noncommutative tori, i.e. non-commutative vector bundles over noncommutative tori. These insights have turned out to be crucial for progress on the proposed objectives. On the one hand, they made it possible to prove the existence of multi-window Gabor frames with good quality (e.g. Schwartz functions), but on the other hand, they provided a way to extend the results of Connes and Rieffel from smooth noncommutative tori to the class of twisted group algebras via the use of methods from Gabor analysis.
Furthermore, we were able to develop a new approach to the K-theory of noncommutative tori and rotation algebras, since projections in noncommutative tori turned out to be generated by tight Gabor atoms. As a first application, I was able to construct new projections in noncommutative tori. Furthermore, these investigations demonstrate that the duality theory of Gabor frames has a natural description in the setting of Morita-Rieffel equivalence of non-commutative tori, in particular that the Wexler-Raz biorthogonality conditions are intrinsically related to projections in non-commutative tori. These investigations also shed new light on the quantum theta functions of Yuri I. Manin, and the well-known projections of Boca. In a two-dimensional setting, non-commutative tori may be viewed as rotation algebras. If one follows the dictionary, one arrives at an operator-algebraic approach to the Walnut representation of Gabor frames, and the conditions on the existence of projections in rotation algebras are related to well-known characterisation of Gabor frames in terms of the Walnut representation. Furthermore, this research demonstrates that the famous Rieffel projections in irrational rotation algebras are actually given by a special class of Gabor frames known as painless non-orthogonal Gabor expansions. Other main results show that a particular Wiener amalgam space provides a natural class of functions leading to projections in rotation algebras and an important result of Kristal and Okoudjou's work on Gabor frames implies the existence of well-localised projections in rotation algebras.
These results provided a way to clarify the relation between Rieffel's work on coupling constants among von Neumann algebras, and the density theorem of Gabor analysis: a proof of the Stone-von Neumann theorem based on methods from time-frequency analysis. The key of the argument is a description of projections in the twisted group algebra of the phase space, which might be considered as in the case of noncommutative tori, where the density of lattices is arbitrarily large. According to Rieffel's findings, this yields results on the structure of noncommutative tori over the adjoint noncommutative tori, which in this case are objects generated by time-frequency shifts over a lattice with an arbitrarily small density.
In addition, these investigations led to the question of when noncommutative tori are Morita-Rieffel equivalent to the continuous functions over the torus, since this is of predominant relevance for the application of Gabor frames to real-world problems. In a short note I was able to show that this is exactly the case if the lattice generating a Gabor frame equals its adjoint, and this is precisely the case if the lattice is symplectic.
Another step towards this objective is to work out the finite-dimensional case, which amounts to the construction of projections in matrix algebras. At the moment I am working out the structures in this setting and it turns out that these are closely connected to algorithms used in wireless communication. Consequently, the Morita-Rieffel equivalence of matrix algebras provides the theoretical framework for many algorithms designed by engineers in signal analysis, e.g. in the transmission of signals via cellular phones.
Finally, we worked on some aspects of noncommutative quantum mechanics in collaboration with two top experts of this theory, Dias and Prata, and with the help of Maurice de Gosson we were able to give a mathematically rigorous formulation. In these works we study noncommutative quantum mechanics from the point of view of deformation quantisation. These investigations have a natural connection to noncommutative geometry and the structure of noncommutative tori, which we want to address in subsequent work.