### Abstract

Let Γ be a countable discrete group. The invariant uniform Roe algebra of G is the C*-subalgebra of its uniform Roe algebra consisting of Γ-invariant elements. We show that Γ has the approximation property if and only if Γ is exact and the invariant uniform Roe algebra has a certain slice map property. This answers a question of J. Zacharias. We also show that characterisations of several properties of Γ in terms of its reduced group C*-algebra also apply to its invariant uniform Roe algebra.

Original language | English |
---|---|

Pages (from-to) | 549-556 |

Number of pages | 8 |

Journal | Journal of Operator Theory |

Volume | 72 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 |

### Keywords

- Approximation property
- Invariant translation approximation property
- Operator approximation property
- Uniform Roe algebra

### ASJC Scopus subject areas

- Algebra and Number Theory

## Fingerprint Dive into the research topics of 'On the invariant uniform Roe algebra'. Together they form a unique fingerprint.

## Cite this

Katsura, T., & Uuye, O. (2014). On the invariant uniform Roe algebra.

*Journal of Operator Theory*,*72*(2), 549-556. https://doi.org/10.7900/jot.2013aug24.2005