Model actions for almost reduced groups on UHF algebras

TL;DR

This paper constructs specific group actions on the universal UHF algebra for groups with certain abelian subgroups, demonstrating these actions have Rokhlin properties and analyzing the resulting crossed products.

Contribution

It introduces a method to construct group actions with Rokhlin properties on UHF algebras for groups with finite index abelian subgroups, and computes their invariants.

Findings

Crossed products are tracially AF with a unique trace

Actions possess Rokhlin property

Elliott invariants computed for abelian groups

Abstract

For any countable discrete group $G$ with a reduced abelian subgroup of finite index, we construct an action $\alpha$ of $G$ on the universal UHF algebra $\Qq$ using an infinite tensor product of permutation representations of $G$ and show that these actions possess some sort of Rokhlin property. The crossed product $\qag$ is then deduced to be tracially AF with a unique tracial state. We also compute the Elliott invariants in the case that $G$ is abelian.