Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A relation between invariant means on Lie groups and invariant means on their discrete subgroups

Author: John R. Grosvenor
Journal: Trans. Amer. Math. Soc. 288 (1985), 813-825
MSC: Primary 43A07; Secondary 22D25, 22E35, 22E40
MathSciNet review: 776406
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Abstract: Let $ G$ be a Lie group, and let $ D$ be a discrete subgroup of $ G$ such that the right coset space $ D\backslash G$ has finite right-invariant volume. We will exhibit an injection of left-invariant means on $ {l^\infty }(D)$ into left-invariant means on the left uniformly continuous bounded functions of $ G$. When $ G$ is an abelian Lie group with finitely many connected components, we also show surjectivity, and when $ G$ is the additive group $ {{\mathbf{R}}^n}$ and $ D$ is $ {{\mathbf{Z}}^n}$, the bijection will explicitly take the form of an integral over the unit cube $ {[0,1]^n}$.

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Keywords: Invariant mean, topological invariant mean
Article copyright: © Copyright 1985 American Mathematical Society