When is an invariant mean the limit of a F{\o}lner net?

John Hopfensperger

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arXiv · 2003.00251

When is an invariant mean the limit of a F{\o}lner net?

John Hopfensperger

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Abstract

Let $G$ be a locally compact amenable group, $TLIM(G)$ the topological left-invariant means on $G$, and $TLIM_0(G)$ the limit points of Folner-nets. I show that $TLIM_0(G) = TLIM(G)$ unless $G$ is $\sigma$-compact non-unimodular, in which case $TLIM_0(G) \neq TLIM(G)$. This improves a 1970 result of Chou and a 2009 result of Hindman and Strauss. I consider the analogous problem for the non-topological left-invariant means, and give a short construction of a net converging to invariance "weakly but not strongly," simplifying the proof of a 2001 result of Rosenblatt and Willis.

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John Hopfensperger. 2020-02-29. When is an invariant mean the limit of a F{\o}lner net?. https://arxiv.org/abs/2003.00251

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