We show spectral invariance for faithful ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-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 Gc\documentclass[12pt]{minimal}
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\begin{document}$$G_c$$\end{document} is C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-unique and symmetric, then the twisted convolution algebra L1(G,c)\documentclass[12pt]{minimal}
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\begin{document}$$L^1 (G,c)$$\end{document} is spectrally invariant in B(H)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {B}}({\mathcal {H}})$$\end{document} for any faithful ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document}-representation of L1(G,c)\documentclass[12pt]{minimal}
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\begin{document}$$L^1 (G,c)$$\end{document} as bounded operators on a Hilbert space H\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}$$\end{document}. As an application of this result we give a proof of the statement that if Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} is a closed cocompact subgroup of the phase space of a locally compact abelian group G′\documentclass[12pt]{minimal}
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\begin{document}$$G'$$\end{document}, and if g is some function in the Feichtinger algebra S0(G′)\documentclass[12pt]{minimal}
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\begin{document}$$S_0 (G')$$\end{document} that generates a Gabor frame for L2(G′)\documentclass[12pt]{minimal}
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\begin{document}$$L^2 (G')$$\end{document} over Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document}, then both the canonical dual atom and the canonical tight atom associated to g are also in S0(G′)\documentclass[12pt]{minimal}
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\begin{document}$$S_0 (G')$$\end{document}. We do this without the use of periodization techniques from Gabor analysis.
