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December 2020 Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters
John Hopfensperger
Rocky Mountain J. Math. 50(6): 2103-2115 (December 2020). DOI: 10.1216/rmj.2020.50.2103


In 1970, Chou showed there are ||=22 topologically invariant means on L(G) for any noncompact, σ-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on L(G) and VN(G) were determined for any locally compact group. Each paper on a new case reached the same conclusion — “the cardinality is as large as possible” — but a unified proof never emerged. In this paper, I show L1(G) and A(G) always contain orthogonal nets converging to invariance. An orthogonal net indexed by Γ has |Γ| accumulation points, where |Γ| is determined by ultrafilter theory.

Among a smattering of other results, I prove Paterson’s conjecture that left and right topologically invariant means on L(G) coincide if and only if G has precompact conjugacy classes.

Finally, I discuss some open problems arising from the study of the sizes of sets of invariant means on groups and semigroups.


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John Hopfensperger. "Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters." Rocky Mountain J. Math. 50 (6) 2103 - 2115, December 2020.


Received: 25 February 2020; Revised: 19 May 2020; Accepted: 26 May 2020; Published: December 2020
First available in Project Euclid: 5 January 2021

Digital Object Identifier: 10.1216/rmj.2020.50.2103

Primary: 20F24 , ‎43A07‎ , 43A30

Keywords: amenable groups , Cardinality , FC groups , Fourier algebras , invariant means , ultrafilters

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


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Vol.50 • No. 6 • December 2020
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