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.

我们展示了一类扭曲卷积代数的忠实\(*\)表示的谱不变性。更准确地说，如果G是一个具有连续 2-cocycle c的局部紧群，其对应的 Mackey 群\(G_c\)是\(C^*\) -唯一且对称的，那么扭曲卷积代数\(L^ 1 (G,c)\)在\({\mathbb {B}}({\mathcal {H}})\) 中对于任何忠实的\(*\) -表示\(L^1 (G ,c)\)作为希尔伯特空间\({\mathcal {H}}\)上的有界运算符。作为这个结果的应用，我们给出一个陈述的证明，如果\(\Delta \)是局部紧阿贝尔群\(G'\)的相空间的封闭协紧子群，如果g是 Feichtinger 代数\(S_0 (G')\)中的某个函数，它会生成一个\(L^2 (G')\)在\(\Delta \) 上的Gabor 框架，那么规范对偶原子和与g相关的规范紧原子也在\(S_0 (G')\) 中。我们在不使用 Gabor 分析中的分期技术的情况下做到这一点。