Covering dimension of Cuntz semigroups (Q2064465)

Inspired by the nuclear dimension of \(C^*\)-algebras, a noncommutative variant of Lebesgue's covering dimension, the authors introduce the notion of covering dimension of Cuntz semigroups. As a motivating example, they show that the covering dimension of \(\mathrm{Lsc}(X, \mathbb N \cup \{\infty\})\), the set of lower-semicontinuous functions from \(X\) to \(\mathbb N \cup \{\infty\}\) is equal to the covering dimension of the compact metrizable space \(X\), i.e., \[ \mathrm{dim}(\mathrm{Lsc}(X,\mathbb N \cup \{\infty\})) = \dim X. \] This dimension theory is well behaved and the authors establish several permanence properties. Regarding the Cuntz semigroup associated to a \(C^*\)-algebra \(A\), the authors show that the covering dimension of \(\mathrm{Cu}(A)\) is bounded by the nuclear dimension of \(A\). In the commutative case, both values agree and they are equal to the dimension of its spectrum. The authors also show that this result holds for the more general class of subhomogeneous \(C^*\)-algebras. Purely infiniteness or real rank zero are conditions implying that the covering dimension of the Cuntz semigroup is equal to 0. The authors provide a partial converse to this latter result if one also assumes unitality and stable rank one. They also show that the covering dimension of the Cuntz semigroup of \(\mathcal{Z}\)-stable \(C^*\)-algebras is at most one. In particular, if we also assume simplicity, then they give the following characterization: \[ \dim ( \mathrm{Cu}(A)) = \begin{cases} 0 & \text{if } A \text{ has real rank zero or } A \text{ is stably projectionless},\\ 1 & \text{otherwise}. \end{cases} \] For Part II, see [\textit{H. Thiel} and \textit{E. Vilalta}, Int. J. Math. 32, No.~14, Article ID 2150100, 27 p. (2021; Zbl 1489.46063)].