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

TL;DR

This paper demonstrates that the unitary Cuntz semigroup provides more detailed classification information for non-simple C*-algebras than traditional invariants, highlighting its importance in the field.

Contribution

It introduces the unitary Cuntz semigroup as a more complete invariant for non-simple C*-algebras and shows it can distinguish algebras that other invariants cannot.

Findings

The unitary Cuntz semigroup contains strictly more information than Cu and K_1 for non-simple C*-algebras.

Two non-simple C*-algebras can have identical Cu and K_1 but differ in their unitary Cuntz semigroups.

The unitary Cuntz semigroup is crucial for the classification of non-simple C*-algebras.

Abstract

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$^*$-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.