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

We study $p$-approximation properties of ${\mathrm{\ell }}^p$ 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 ${\mathrm{\ell }}^p$ uniform Roe algebra $B_u^p(X)$ for every $p\in (1,\mathrm{\infty })$, while $B_u^1(X)$ 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 ${\mathrm{\ell }}^p$ uniform Roe algebras with coefficients in $p$-operator spaces.