具有强次指数增长群的谱不变性和盖弗雷正则性
Spectral Invariance and Gevrey Regularity for Groups with strongly subexponential growth
AI总结：
研究局部紧群上卷积算子的谱不变性和盖弗雷正则性，利用次指数权重，对特定增长群建立相关结果，构造盖弗雷 - 贝林算子代数并证明其性质及应用，还发展了有限生成群对的相关理论。
摘要：
我们研究了配备局部有界长度函数$\ell$的局部紧群上，核在合适加权函数空间中的卷积算子的谱不变性和盖弗雷正则性。主要分析尺度由次指数权重给出。对于体积增长被$e^{R^\gamma}$（其中$0<\gamma<1$）上界限制的群，我们建立了紧支集函数的谱比较结果。对于紧支集埃尔米特函数，我们证明了在对称$q$ - 伪函数$*$ - 代数、加权和未加权群代数以及全群和约化群$C^*$ - 代数上谱半径的不变性。对于指数至多为$\beta$的满足强次指数增长的幺模群，我们在单位化的$q$ - 伪函数代数内构造了一个盖弗雷 - 贝林算子代数。我们证明这个代数是逆封闭的，并且它的包含在拓扑$K$ - 理论中诱导同构。逆封闭性定理可以看作是一个定量的盖弗雷型非交换维纳引理。作为应用，我们表明，只要核在相应加权盖弗雷 - 贝林空间中的卷积算子在单位化的$q$ - 伪函数代数中可逆，那么它的逆属于同一个盖弗雷 - 贝林算子代数并且满足明确的盖弗雷半范数估计。我们还使用施赖埃尔图长度和拟正则表示发展了有限生成群对的相关理论。这为群对提供了次指数类似的快速衰减，当子群是正规子群时，它简化为商上的通常理论。该框架可以应用于中间增长的例子，包括格里戈丘克群，并且在与多项式增长群的乘积和紧扩张下是稳定的。
英文摘要：
We study spectral invariance and Gevrey regularity for convolution operators with kernels in suitable weighted function spaces on locally compact groups equipped with a locally bounded length function $\ell$. The main analytic scale is given by the subexponential weights. For groups whose volume growth is bounded above by $e^{R^γ}$ for some $0<γ<1$, we establish spectral comparison result for compactly supported functions. For compactly supported Hermitian functions, we prove spectral radius invariance across the symmetric $q$-pseudofunction $*$-algebra, the weighted and unweighted group algebras, and the full and reduced group $C^*$-algebras. For unimodular groups satisfying strong subexponential growth of exponent at most $β$, we construct a Gevrey-Beurling operator algebra inside the unitized $q$-pseudofunction algebra. We prove that this algebra is inverse-closed and that its inclusion induces an isomorphism in topological $K$-theory. The inverse-closedness theorem may be viewed as a quantitative Gevrey-type noncommutative Wiener lemma. As an application, we show that whenever a convolution operators with kernels in the corresponding weighted Gevrey-Beurling space is invertible in the unitized $q$-pseudofunction algebra, then its inverse belongs to the same Gevrey-Beurling operator algebra and satisfies explicit Gevrey seminorm estimates. We also develop a relative theory for pairs of finitely generated groups using Schreier graph lengths and quasi-regular representations. This provides a subexponential analogue of rapid decay for group pairs, when subgroup is normal, it reduces to the usual theory on the quotient. The framework can apply to intermediate-growth examples, including the Grigorchuk group, and is stable under products with polynomial growth groups and under compact extensions.
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