Counting topologically invariant means on $L_\infty(G)$ and $VN(G)$ with ultrafilters

Abstract

In 1970, Chou showed there are $\left|{\mathbb{N}}^{\ast }\right|=2^{2^{\mathbb{N}}}$ topologically invariant means on $L_{\infty }\left(G\right)$ for any noncompact, $\sigma$-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on $L_{\infty }\left(G\right)$ and $VN\left(G\right)$ 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 $L_1\left(G\right)$ and $A\left(G\right)$ always contain orthogonal nets converging to invariance. An orthogonal net indexed by $\mathrm{\Gamma }$ has $\left|{\mathrm{\Gamma }}^{\ast }\right|$ accumulation points, where $\left|{\mathrm{\Gamma }}^{\ast }\right|$ is determined by ultrafilter theory.

Among a smattering of other results, I prove Paterson’s conjecture that left and right topologically invariant means on $L_{\infty }\left(G\right)$ 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.