Topological Invariant Means on Locally Compact Groups

TL;DR

This thesis explores the structure and properties of topological invariant means on amenable locally compact groups, revealing their diversity, uniqueness, and symmetry conditions through advanced functional analysis and topology techniques.

Contribution

It establishes new results on the existence, uniqueness, and symmetry of topological invariant means, including their relation to F ext{"o}lner nets and lattice subgroups.

Findings

Existence of F ext{"o}lner nets with many accumulation points

Uniqueness of accumulation points for unimodular groups

Bijection between invariant means on subgroups and quotients

Abstract

Suppose $G$ is an amenable locally compact group. If $\{F_\gamma\} = \{F_\gamma\}_{\gamma\in\Gamma}$ is a F\o{}lner net for $G$, associate it with the net $\{\chi_{F_\gamma} / |F_\gamma|\} \subset L_1(G) \subset L_\infty^*(G)$. Thus, every accumulation point of $\{F_\gamma\}$ is a topological left-invariant mean on $G$. The following are examples of results proved in the present thesis: (1) There exists a F\o{}lner net which has as its accumulation points a set of $2^{2^{\kappa}}$ distinct topological left-invariant means on $G$, where $\kappa$ is the smallest cardinality of a covering of $G$ by compact subsets. (2) If $G$ is unimodular and $\mu$ is a topological left-invariant mean on $G$, there exists a F\o{}lner net which has $\mu$ as its unique accumulation point. (3) Suppose $L \subset G$ is a lattice subgroup. There is a natural bijection of the left-invariant means on $L$ with the topological left-invariant means on $G$ if and only if $G/L$ is compact. (4) Every topological left-invariant mean on $G$ is also topological right-invariant if and only if $G$ has precompact conjugacy classes. These results lie at the intersection of functional analysis with general topology. Problems in this area can often be solved with standard tools when $G$ is $\sigma$-compact or metrizable, but require more interesting arguments in the general case.