On the invariant uniform Roe algebra

Let $Γ$ be a countable discrete group. We show that $Γ$ has the approximation property if and only if $Γ$ is exact and for any operator space $S \subseteq \K(H)$ we have $\Cu(Γ)^Γ \otimes S = (\Cu(Γ) \otimes S)^Γ$, where $\Cu(Γ)$ is the uniform Roe algebra with the right adjoint $Γ$-action. This answers a question of J. Zacharias. We also show that characterisations of several properties of $Γ$ in terms of the reduced group \cast-algebra apply to the invariant uniform Roe algebra $\Cu(Γ)^Γ$.

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