酉表示下随机游动的谱间隙性质
A spectral gap property for random walks under unitary representations
摘要：
设$G$是局部紧群，$μ$是$G,$上的概率测度，且不假定它关于Haar测度是绝对连续的.给定一个酉表示$(π, \cal H)$的$G,$，我们研究的频谱性质的运营商$π(μ)$作用于$\cal H.$假设$μ$是适应和平凡的表示$1_G$不弱包含在张量积$π\otimes \barπ.$我们表明，$π(μ)$有一个频谱间隙，即，对于谱半径$r_{\rm spec}(π(μ))$的$π(μ),$，我们有$r_{\rm spec}(π(μ))<1.$这提供了一个共同的推广几个以前已知的结果。另一个结果是，如果$G$具有Kazhdan性质（T），则对于没有有限维子表示的$G$的每个酉表示$π$，$r_{\rm spec}(π(μ))<1$。此外，我们给出了新的例子，所谓的身份排除群。
英文摘要：
Let $G$ be a locally compact group and $μ$ a probability measure on $G,$ which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $(π, \cal H)$ of $G,$ we study spectral properties of the operator $π(μ)$ acting on $\cal H.$ Assume that $μ$ is adapted and that the trivial representation $1_G$ is not weakly contained in the tensor product $π\otimes \barπ.$ We show that $π(μ)$ has a spectral gap, that is, for the spectral radius $r_{\rm spec}(π(μ))$ of $π(μ),$ we have $r_{\rm spec}(π(μ))<1.$ This provides a common generalization of several previously known results. Another consequence is that, if $G$ has Kazhdan's Property (T), then $r_{\rm spec}(π(μ))<1$ for every unitary representation $π$ of $G$ without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.
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