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 B(H) for any faithful * -representation of L 1 (G, c) as bounded operators on a Hilbert space H. As an application of this result we give a proof of the statement that if ∆ 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 ∆, 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.