Abstract: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.
From: Yeong Chyuan Chung [view email]
[v1]
Thu, 25 Jun 2026 16:17:02 UTC (29 KB)
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