Unitary Cuntz semigroups of ideals and quotients

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${}^{\ast }$-algebra of stable rank one $A$, the assignment $I⟼$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${}_1(A)\mathrm{/}$Cu${}_1(I)\simeq$ Cu${}_1(A\mathrm{/}I)$ 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${}^{\ast }$-algebra of stable rank one.