In this paper, we characterize when the $\ell^p$ uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for $p\in [1,\infty)$. Moreover, we show that the $\ell^p$ uniform Roe algebra is a (non-sequential) spatial $L^p$ AF algebra in the sense of Phillips and Viola if and only if the underlying metric space has asymptotic dimension zero.

We also consider the ordered $K_0$ groups of $\ell^p$ uniform Roe algebras for metric spaces with low asymptotic dimension, showing that (1) the ordered $K_0$ group is trivial when the metric space is non-amenable and has asymptotic dimension at most one, and (2) when the metric space is a countable locally finite group, the (ordered) $K_0$ group is a complete invariant for the (bijective) coarse equivalence class of the underlying locally finite group. It happens that in both cases the ordered $K_0$ group does not depend on $p\in [1,\infty)$.

在本文中，我们刻画了具有有界几何的度量空间的 $\ell^p$ 统一 Roe 代数何时是（稳定）有限的，以及何时在 $p\in [1,\infty)$ 的标准形式中适当地无限. 此外，我们证明 $\ell^p$ 统一 Roe 代数是 Phillips 和 Viola 意义上的（非序列）空间 $L^p$ AF 代数，当且仅当基础度量空间具有渐近维数为零。

我们还考虑了具有低渐近维数的度量空间的 $\ell^p$ 统一 Roe 代数的有序 $K_0$ 群，表明（1）当度量空间不适合并且具有渐近维最多为 1，并且 (2) 当度量空间是可数局部有限群时，（有序）$K_0$ 群是底层局部有限群的（双射）粗等价类的完全不变量。碰巧在这两种情况下，有序的 $K_0$ 组都不依赖于 $p\in [1,\infty)$。