The inversion characterizations of $C(\mathcal {L})$ for a locale $\mathcal {L}$

Abstract

The category $\mathbf {W}$ is comprised of archimedean $\ell $-groups $G$ with distinguished weak unit and unit-preserving $\ell $-group homomorphisms. For $G \in \mathbf {W}$ there is the canonical Yosida representation $G \leq D(\mathcal {Y}G)$ as extended-real valued functions, with the unit $e_G$ represented as the constant $1$ function. We define $a \in G$ to be kernel-maximal if the $\mathbf {W}$-kernel generated by $a$ in $G$ is all of $G$, and we define $a$ to be Yosida invertible if there is some $b \in G$ with $ab = e_G$ in the partial multiplication inherited from $D(\mathcal {Y}G)$. The main theorem is that $G$ is isomorphic to $C(\mathcal {L})$ for some (identifiable) locale $\mathcal {L}$ iff $G$ is divisible, uniformly complete, and every kernel-maximal element of $G$ is Yosida invertible.