Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Eduardo Scarparo

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Let $(X,T,\mu)$ be a Cantor minimal system 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 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.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 26, Number 3 (2019), 469-479.

First available in Project Euclid: 17 September 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]

group $C^*$-algebra real rank zero topological full group


Scarparo, Eduardo. On the $C^*$-algebra generated by the Koopman representation of a topological full group. Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 3, 469--479. doi:10.36045/bbms/1568685659. https://projecteuclid.org/euclid.bbms/1568685659

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