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A BIJECTION OF INVARIANT MEANS ON AN AMENABLE GROUP WITH THOSE ON A LATTICE SUBGROUP

Published online by Cambridge University Press:  18 January 2021

JOHN HOPFENSPERGER*
Affiliation:
Department of Mathematics, University at Buffalo, Buffalo, NY14260-2900, USA

Abstract

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.

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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References

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