Diagonal limits of finite alternating groups: confined subgroups, ideals, and positive definite functions

Abstract

A non-finitary group $G$ is said to be an $\mathrm{\small L}\mathfrak{A}$-group if it is a direct limit of finite alternating groups $G_i=\operatorname{Alt}({\Omega_i})$ $(i\!\in\!I)$ such that each $G_i$ has only trivial or natural orbits on the sets $\Omega_j$ $(j\! \gt \!i)$. We determine the confined subgroups of $\mathrm{\small L}\mathfrak{A}$-groups and relate them naturally to the ideals in the group algebra $\mathbb{K} G$ over any field $\mathbb{K}$ of characteristic zero. Moreover, we show that the non-trivial ideals in $\mathbb{C} G$ can be related to normalized positive definite class functions $f\colon G\to\mathbb{C}$ if and only if the number of $G_i$-orbits in $\Omega_j$ $(j\! \gt \!i)$ is asymptotically a linear function of $|\Omega_j|$ for all $i$.