### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Operator Theory*,

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

**On the invariant uniform Roe algebra.** / Katsura, Takeshi; Uuye, Otgonbayar.

Research output: Contribution to journal › Article

*Journal of Operator Theory*, vol. 72, no. 2, pp. 549-556. https://doi.org/10.7900/jot.2013aug24.2005

}

TY - JOUR

T1 - On the invariant uniform Roe algebra

AU - Katsura, Takeshi

AU - Uuye, Otgonbayar

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Approximation property

KW - Invariant translation approximation property

KW - Operator approximation property

KW - Uniform Roe algebra

UR - http://www.scopus.com/inward/record.url?scp=84920601406&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84920601406&partnerID=8YFLogxK

U2 - 10.7900/jot.2013aug24.2005

DO - 10.7900/jot.2013aug24.2005

M3 - Article

AN - SCOPUS:84920601406

VL - 72

SP - 549

EP - 556

JO - Journal of Operator Theory

JF - Journal of Operator Theory

SN - 0379-4024

IS - 2

ER -