论文标题
准计数的逆半群作为度量空间，以及局部有限逆半群的均匀ROE代数
Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups
论文作者
论文摘要
考虑到任何准计数，尤其是任何可计数的反向半群$ s $，我们引入了一种方法，以配备$ s $的适当且正确的次级扩展指标。这概括了离散可数组的适当，正确不变的指标的概念。这种度量标准被证明是独特的，这是半群的双重粗糙等效性，因此基本上仅取决于$ s $。这使我们能够明确定义$ s $的统一ROE代数，我们证明可以将其实现为$ \ ell^\ infty（s）$和$ s $的规范交叉产品。我们将这些指标与Hausdorffétalegropoids的类似指标联系起来。 使用此设置，我们研究了那些具有渐近维度$ 0 $的反向半群。概括了针对组已知的结果，我们表明这些恰恰是局部有限的反向半群，并且进一步的特征是具有强烈的准二角形均匀的ROE代数。我们表明，与集体案例不同，具有有限的统一ROE代数严格较弱，其特征是$ s $是本地$ \ MATHCAL L $ -FINITE，并且等效地稀疏为度量空间。
Given any quasi-countable, in particular any countable inverse semigroup $S$, we introduce a way to equip $S$ with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on $S$. This allows us to unambiguously define the uniform Roe algebra of $S$, which we prove can be realized as a canonical crossed product of $\ell^\infty(S)$ and $S$. We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension $0$. Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups, and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by $S$ being locally $\mathcal L$-finite, and equivalently sparse as a metric space.
