Perforation conditions and almost algebraic order in Cuntz semigroups

TL;DR

This paper investigates how the Cuntz semigroup of a C*-algebra behaves under tensoring with the Jiang-Su algebra, revealing conditions under which certain algebraic and order properties are preserved or reflected.

Contribution

It provides a detailed analysis of the passage of properties from the Cuntz semigroup of a C*-algebra to its tensor product with the Jiang-Su algebra within the abstract semigroup category.

Findings

Tensoring with  preserves certain order properties of the Cuntz semigroup.

For real rank zero algebras without elementary subquotients, the tensor product has a dense set of projections.

Almost unperforation of projections characterizes tensoring with  as inert at the semigroup level.

Abstract

For a C$^*$-algebra $A$, it is an important problem to determine the Cuntz semigroup $\mathrm{Cu}(A\otimes\mathcal{Z})$ in terms of $\mathrm{Cu}(A)$. We approach this problem from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups, by analysing the passage of significant properties from $\mathrm{Cu}(A)$ to $\mathrm{Cu}(A)\otimes_\mathrm{Cu}\mathrm{Cu}(\mathcal{Z})$. We describe the effect of the natural map $\mathrm{Cu}(A)\to\mathrm{Cu}(A)\otimes_\mathrm{Cu}\mathrm{Cu}(\mathcal{Z})$ in the order of $\mathrm{Cu}(A)$, and show that, if $A$ has real rank zero and no elementary subquotients, $\mathrm{Cu}(A)\otimes_\mathrm{Cu}\mathrm{Cu}(\mathcal{Z})$ enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, nonelementary, real rank zero and stable rank one situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with $\mathcal{Z}$ is inert at the level of the Cuntz semigroup.