## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 17 September 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1568685659

**Digital Object Identifier**

doi:10.36045/bbms/1568685659

**Mathematical Reviews number (MathSciNet)**

MR4007610

**Subjects**

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

**Keywords**

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

#### Citation

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