In 1970, Chou showed there are topologically invariant means on for any noncompact, -compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on and 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 and always contain orthogonal nets converging to invariance. An orthogonal net indexed by has accumulation points, where is determined by ultrafilter theory.
Among a smattering of other results, I prove Paterson’s conjecture that left and right topologically invariant means on coincide if and only if 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.
"Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters." Rocky Mountain J. Math. 50 (6) 2103 - 2115, December 2020. https://doi.org/10.1216/rmj.2020.50.2103