拟不变性群与庞特里亚金对偶
Groups of quasi-invariance and the Pontryagin duality
摘要：
波兰群$G$称为拟不变性群或QI群，如果存在局部紧群$X$和$X$上的概率测度$μ$，使得1）存在$G$到$X$的连续单态，2）对于每个$g\in X$，$g\in G$和移位$μ_g$等价于$μ$或$g\not\in G$，并且$μ_g$与$μ$正交。证明了$G$是$X$的$σ$ -紧子集.证明了存在$\ell^2$模离散子群的商群$\mathbb{T}^H_2$，该离散子群是波兰单素非局部拟凸（因而是非自反的）路径连通的QI-群，且使得$\mathbb{T}^H_2$的双对偶不是QI-群.还证明了QI-群的双对偶群可能不是$X$的饱和子群。
英文摘要：
A Polish group $G$ is called a group of quasi-invariance or a QI-group, if there exist a locally compact group $X$ and a probability measure $μ$ on $X$ such that 1) there exists a continuous monomorphism of $G$ to $X$, and 2) for each $g\in X$ either $g\in G$ and the shift $μ_g$ is equivalent to $μ$ or $g\not\in G$ and $μ_g$ is orthogonal to $μ$. It is proved that $G$ is a $σ$-compact subset of $X$. We show that there exists a quotient group $\mathbb{T}^H_2$ of $\ell^2$ modulo a discrete subgroup which is a Polish monothetic non locally quasi-convex (and hence nonreflexive) pathwise connected QI-group, and such that the bidual of $\mathbb{T}^H_2$ is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of $X$.
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