MOTIF · Modern Methods of Operator Algebras for Time-Frequency Analysis

The project is on the interface between time-frequency analysis and operator algebras. In the last decade the relevance of operator algebras in time-frequency analysis and Gabor analysis has evolved mainly due to the contributions of Groechenig, Janssen and their collaborators. The operator algebras that have appeared in these works are twisted group C*-algebras or more generally crossed products. In the present project the proposer wants to explore the deeper structure of crossed products to gain some new insight about time-varying filters and Gabor frames. The methods involved in these investigations are modern methods from operator algebras such as groups of automorphisms, projective modules over twisted group C*-algebras and the K-theory of operator algebras. The proposer will invoke deep results of Arveson, Connes, Elliott, Evans, Haagerup, Pimsner, Rieffel and Voiculescu obtained in the study of noncommutative tori and irrational rotation algebras. Another aspect of the proposal deals with varying the lattice of a Gabor frame and the approximation of a continuous Gabor frame by finite-dimensional Gabor frames. The work of Rieffel on strict deformation quantization, continuous fields of C*-algebras and the notion of quantum Gromov-Hausdorff distance will provide the correct operator algebraic setting for these topics. Coming from NuHAG and EUCETIFA the candidate is at one of the sources of research in time-frequency analysis. The choice of the outgoing host institution UC Berkeley is dictated by the concentration of operator analysts there. In particular the established scientific contacts to Rieffel will be very valuable. After the return to Europe the insight obtained will be further developed in cooperation with the researchers there, and the consequences for applications (for the study of time-variant systems etc.) will be exploited.