论文标题
$*$的光谱不变性 - 扭曲卷积代数的表示，并在Gabor分析中申请
Spectral invariance of $*$-representations of twisted convolution algebras with applications in Gabor analysis
论文作者
论文摘要
我们显示了信徒$*$的光谱不变性 - 一类扭曲卷积代数的表示。更准确地说，如果$ g $是本地紧凑的群体，则具有连续$ 2 $ -COCYCLE $ C $，相应的Mackey组$ g_c $是$ c^*$ - 独特而对称的，那么扭曲的卷积代数。 $*$ - $ l^1（g，c）$的表示，作为Hilbert Space $ \ Mathcal {H} $的有限运算符。作为此结果的应用，我们给出了以下声明的证明：如果$δ$是封闭的cocoCompact子组，则是局部紧凑的Abelian $ g'$的相位空间，并且如果$ G $在Feichtinger代数$ s_0（g'）中具有某种功能到$ g $也在$ s_0（g'）$中。我们在不利用Gabor分析的周期技术的情况下这样做。
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 $Δ$ 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 $Δ$, 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.
