Unitary Cuntz semigroups of ideals and quotients

Abstract

We define a notion of ideal for objects in the category of abstract unitary Cuntz semigroups introduced in [3] and termed Cu$^\sim$. We show that the set of ideals of a Cu$^\sim$-semigroup has a complete lattice structure. In fact, we prove that for any C$^*$-algebra of stable rank one $A$, the assignment $I\longmapsto$Cu$_1(I)$ defines a complete lattice isomorphism between the set of ideals of $A$ and the set of ideals of its unitary Cuntz semigroup Cu$_1(A)$. Further, we introduce a notion of quotients and exactness for the (non abelian) category Cu$^\sim$. We show that Cu$_1(A)/$Cu$_1(I)\simeq$ Cu$_1(A/I)$ for any ideal $I$ in $A$ and that the functor Cu$_1$ is exact. Finally, we link a Cu$^\sim$-semigroup with the Cu-semigroup of its positive elements and the abelian group of its maximal elements in a split-exact sequence. This result allows us to extract additional information that lies within the unitary Cuntz semigroup of a C$^*$-algebra of stable rank one.