On some $p$-approximation properties of exact discrete groups and $\ell^p$ uniform Roe algebras

We study $p$-approximation properties of $\ell^p$ uniform Roe algebras and their connections to coarse geometry and group theory.

For a discrete metric space $X$ with bounded geometry, we prove that property A implies $p$-nuclearity of the $\ell^p$ uniform Roe algebra $B^p_u(X)$ for every $p\in(1,\infty)$, while $B^1_u(X)$ is always 1-nuclear.

We introduce the $p$-invariant translation approximation property ($p$-ITAP) for discrete groups, generalizing the 2-ITAP of Roe.

We also introduce the $p$-operator ITAP.

For exact groups, we show that the $p$-operator ITAP is equivalent to the $p$-approximation property of An-Lee-Ruan.

We also characterize exactness of discrete groups in terms of their $\ell^p$ uniform Roe algebras with coefficients in $p$-operator spaces.