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

Part of: Abstract harmonic analysis Noncompact transformation groups

Published online by Cambridge University Press:  18 January 2021

JOHN HOPFENSPERGER Open the ORCID record for JOHN HOPFENSPERGER [Opens in a new window]
JOHN HOPFENSPERGER*
Affiliation:
Department of Mathematics, University at Buffalo, Buffalo, NY14260-2900, USA
*
e-mail: [email protected]
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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.

Keywords

amenable groupsFølner netsinvariant meanslattices

MSC classification

Primary: 43A07: Means on groups, semigroups, etc.; amenable groups
Secondary: 22F30: Homogeneous spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 302 - 307
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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References

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