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.
- Approximation property
- Invariant translation approximation property
- Operator approximation property
- Uniform Roe algebra
ASJC Scopus subject areas
- Algebra and Number Theory