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

Abstract

This paper argues that the unitary Cuntz semigroup, introduced in [10] and termed Cu1, contains crucial information regarding the classification of non-simple C⁎-algebras. We exhibit two non-simple C⁎-algebras that agree on their K1-groups and their Cuntz semigroups (termed Cu) 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⁎-algebras than Cu and K1 alone.