# On the $C^*$-algebra generated by the Koopman representation of a topological full group

## TL;DR

This paper investigates the structure of the $C^*$-algebra generated by the Koopman representation of a topological full group associated with a Cantor minimal system, revealing its relation to crossed product algebras and properties of the group.

## Contribution

It establishes the equality of certain $C^*$-algebras, characterizes the kernel of a specific character, and links the properties of the group to the real rank of its $C^*$-algebra.

## Key findings

- The $C^*$-algebra $C^*_ ext{pi}([[T]])$ equals $C^*_ ext{pi}([[T]]')$.
- The kernel of the character $ au$ is a hereditary subalgebra stably isomorphic to $C(X)
times bZ$.
- If $G$ is finitely generated, elementary amenable, and $C^*(G)$ has real rank zero, then $G$ is finite.

## Abstract

Let $(X,T,\mu)$ be a Cantor minimal sytem and $[[T]]$ the associated topological full group. We analyze $C^*_\pi([[T]])$, where $\pi$ is the Koopman representation attached to the action of $[[T]]$ on $(X,\mu)$. Specifically, we show that $C^*_\pi([[T]])=C^*_\pi([[T]]')$ and that the kernel of the character $\tau$ on $C^*_\pi([[T]])$ coming from weak containment of the trivial representation is a hereditary $C^*$-subalgebra of $C(X)\rtimes\mathbb{Z}$. Consequently, $\ker\tau$ is stably isomorphic to $C(X)\rtimes\mathbb{Z}$, and $C^*_\pi([[T]]')$ is not AF. We also prove that if $G$ is a finitely generated, elementary amenable group and $C^ *(G)$ has real rank zero, then $G$ is finite.