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

Abstract

This paper argues that the unitary Cuntz semigroup, introduced in [9] and termed Cu 1 , contains crucial information regarding the classification of non-simple C * -algebras. We exhibit two (non-simple) C * -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 * -algebras than Cu and K 1 alone.