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

## TL;DR

This paper investigates the size and structure of topologically invariant means on function spaces related to locally compact groups, proving new results about their cardinality and convergence properties using ultrafilter theory.

## Contribution

It introduces a unified approach to analyze invariant means on $L_ ext{infty}(G)$ and $VN(G)$, proving the existence of orthogonal nets converging to invariance and confirming Paterson's conjecture.

## Key findings

- Cardinality of invariant means is maximally large, as determined by ultrafilter theory.
- Existence of orthogonal nets in $L_1(G)$ and $A(G)$ converging to invariance.
- Proof of Paterson's conjecture relating invariant means to conjugacy class precompactness.

## 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.