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

Abstract

In 1970, Chou showed there are $|\mathbb{N}^*| = 2^{2^\mathbb{N}}$ topologically invariant means on $L_\infty(G)$ for any noncompact, $\sigma$-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on $L_\infty(G)$ 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 $L_1(G)$ and $A(G)$ always contain orthogonal nets converging to invariance. An orthogonal net indexed by $\Gamma$ has $|\Gamma^*|$ accumulation points, where $|\Gamma^*|$ is determined by ultrafilter theory. Among a smattering of other results, I prove Paterson's conjecture that left and right topologically invariant means on $L_\infty(G)$ coincide iff $G$ has precompact conjugacy classes.