From signal analysis and coherent states to projective modules over non-commutative tori
Seminar Room 2, Newton Institute Gatehouse
In the present talk we want to gap the bridge between Gabor analysis and operator algebras. Gabor analysis is a new field of signal analysis, like wavelets, which is closely related with harmonic analysis over the Heisenberg group. We recall Gabor's original idea to expand signals by samples of coherent states. The main aim is the discussion how an applied problem on good window classes for Gabor frames leads to the construction of projective modules over non-commutative tori. These investigations will unreveal that Hilbert modules over non-commutative tori are the natural operator algebraic framework for Gabor analysis. As a consequence we extend Rieffel's and Connes's construction of projective modules over non-commutative tori to other classes of spectral invariant subalgebras of twisted group algebras by the replacement of Schwartz's space by certain modulation spaces. Furthermore we present new results on the structure of projective modules which arose from properties of Gabor frames.