Perforation conditions and almost algebraic order in Cuntz semigroups

Abstract

For a C${}^{\ast }$-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.

30 pages; minor bugs fixed. This paper has been accepted for publication in Proceedings of the Royal Society of Edinburgh Section A Mathematics and will appear in a revised form subsequent to editorial input by the ICMS/Royal Soc. of Edinburgh. Material on these pages is copyright Cambridge University Press. http://journals.cambridge.org/action/displayJournal?jid=PRM