Topological Invariant Means on Locally Compact Groups

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.