Suppose G is an amenable locally compact group with lattice subgroup $\Gamma $ . Grosvenor [‘A relation between invariant means on Lie groups and invariant means on their discrete subgroups’, Trans. Amer. Math. Soc.288(2) (1985), 813–825] showed that there is a natural affine injection $\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that $\iota $ is a surjection essentially in the case $G={\mathbb R}^d$ , $\Gamma ={\mathbb Z}^d$ . In the present paper it is shown that $\iota $ is a surjection if and only if $G/\Gamma $ is compact.

中文翻译：

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不变量均值的双射与格子群上的均值

认为G是具有格子群的服从局部紧群$\伽马$. Grosvenor ['李群上的不变均值与其离散子群上的不变均值之间的关系'，反式。阿米尔。数学。社会党。288(2) (1985), 813-825] 表明存在天然仿射注射$\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$然后$\iota $在这种情况下本质上是一个凸出$G={\mathbb R}^d$,$\Gamma ={\mathbb Z}^d$. 在本论文中表明$\iota $当且仅当$G/\伽玛 $紧凑。