The unitary Cuntz semigroup on the classification of non-simple C*-algebras

This paper argues that the unitary Cuntz semigroup, introduced in [10] and termed Cu${}_1$, contains crucial information regarding the classification of non-simple C${}^{\ast }$-algebras. We exhibit two (non-simple) C${}^{\ast }$-algebras that agree on their Cuntz semigroups, termed Cu, and their K${}_1$-groups and yet disagree at level of their unitary Cuntz semigroups. In the process, we establish that the unitary Cuntz semigroup contains rigorously more information about non-simple C${}^{\ast }$-algebras than Cu and K${}_1$ alone.