论文标题
局部紧凑型组动作差异的通用下限
Universal lower bounds for the discrepancies of actions of a locally compact group
论文作者
论文摘要
我们证明了在概率空间上局部紧凑型组的量度保留作用的差异（即平均运算符的光谱差距的尺寸）的通用下限。例如，局部紧凑的hausdorff unimodular $ g $，通过衡量衡量的变换，在紧凑的原子概率空间$（x，ν）$上连续行动，并在$ν$中构成了$ gx_0 $ gx_0 $ gx_0 $ gx_0 $ gx_0 $ g x $ n of $ c sotiltime compitize and compilter（i.g _ come.g_ come.g _0} $ g _0} $ g _ _0} $ g $上的鲍勒度量$ $ $满足$$ \ |π_0（μ）\ | _ {2 \ to 2} \ geq \ geq \ | | |} $ g $。我们证明，下限为离散组的量度保留行为的差异概括了通用的下限。许多示例表明，从离散组到本地紧凑的组的概括需要对该动作进行一些其他假设（由于Margulis，我们详细介绍了具有光谱差距的可及组的作用的一些例子）。 Kazhdan和Zimmer的众所周知的例子和结果表明，在同质空间上，谎言组的某些动作的差异与我们证明的通用下限完全符合。
We prove universal lower bounds for discrepancies (i.e. sizes of spectral gaps of averaging operators) of measure-preserving actions of a locally compact group on probability spaces. For example, a locally compact Hausdorff unimodular group $G$, acting continuously, by measure-preserving transformations, on a compact atomless probability space $(X,ν)$, with an orbit $Gx_0$ of measure zero, contained in the support of $ν$, and with compact stabilizer (i.e. $G_{x_0}$ is compact) has the following property: any finite positive regular Borel measure $μ$ on $G$ satisfies $$\|π_0(μ)\|_{2\to 2}\geq \|λ_G(μ)\|_{2\to 2},$$ where $π_0$ denotes the Koopman representation of $G$, defined by the given action, and $λ_G$ denotes the left-regular representation of $G$. The lower bounds we prove generalize the universal lower bounds for the discrepancies of measure-preserving actions of a discrete group. Many examples show that the generalization from discrete groups to locally compact groups requires some additional hypothesis on the action (we detail some examples of actions of amenable groups with a spectral gap, due to Margulis). Well-known examples and results of Kazhdan and Zimmer show that the discrepancies of some actions of Lie groups on homogeneous spaces match exactly the universal lower bounds we prove.
