Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

Abstract

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.

Figures (2)

• Figure 1: The two distances $d_1$ and $d_2$ when $G=\mathbb Z_2$ in Example \ref{['ex:need-subinvariance']}
• Figure 2: The two distances $d_1$ and $d_2$ when $G=\mathbb Z_2$ in Example \ref{['ex:nonfl']}

Theorems & Definitions (93)

• Theorem 1: see Theorem \ref{['thm:metric']}
• Theorem 2: see \ref{['thm:wobbling-inv-sem']}
• Theorem 3: see Theorem \ref{['thm:roealg:unique']}
• Theorem 4: see Theorem \ref{['thm:asymdim-zero']}
• Theorem 5: see Theorem \ref{['thm:boxspace']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5