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Let $(X,T,\mu)$ be a Cantor minimal sytem and $[[T]]$ the associatedtopological full group. We analyze $C^*_\pi([[T]])$, where $\pi$ is the Koopmanrepresentation attached to the action of $[[T]]$ on $(X,\mu)$.  Specifically, we show that $C^*_\pi([[T]])=C^*_\pi([[T]]')$ and that thekernel of the character $\tau$ on $C^*_\pi([[T]])$ coming from weak containmentof 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 groupand $C^ *(G)$ has real rank zero, then $G$ is finite.

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