Spectral Invariance of *-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

Abstract

We show spectral invariance for faithful *-representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group G_c is C^*-unique and symmetric, then the twisted convolution algebra L^1 (G,c) is spectrally invariant in {mathbb {B}}({mathcal {H}}) for any faithful *-representation of L^1 (G,c) as bounded operators on a Hilbert space {mathcal {H}}. As an application of this result we give a proof of the statement that if Delta is a closed cocompact subgroup of the phase space of a locally compact abelian group G', and if g is some function in the Feichtinger algebra S_0 (G') that generates a Gabor frame for L^2 (G') over Delta , then both the canonical dual atom and the canonical tight atom associated to g are also in S_0 (G'). We do this without the use of periodization techniques from Gabor analysis.