A unitary Cuntz semigroup for C*-algebras of stable rank one

TL;DR

This paper introduces the Cu$_1$-semigroup, a new invariant combining the Cuntz semigroup and K$_1$-group information for C*-algebras of stable rank one, providing a unified and computable tool.

Contribution

The paper defines the Cu$_1$-semigroup as a new invariant that merges Cuntz semigroup and K$_1$ data, and shows it can recover existing invariants functorially.

Findings

Cu$_1$-semigroup is a well-defined continuous functor

Computed Cu$_1$ for specific C*-algebras

Can recover Cu, K$_1$, and K$_*$ from Cu$_1$

Abstract

We introduce a new invariant for C*-algebras of stable rank one that merges the Cuntz semigroup information together with the K$_1$-group information. This semigroup, termed the Cu$_1$-semigroup, is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given C*-algebra together with a unitary element of the unitization of the hereditary subalgebra generated by the given positive element. We show that the Cu$_1$-semigroup is a well-defined continuous functor from the category of C*-algebras of stable rank one to a suitable codomain category that we write Cu$^\sim$. Furthermore, we compute the Cu$_1$-semigroup of some specific classes of C*-algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, K$_1$ and K$_*:=$K$_0$$\oplus$ K$_1$ from Cu$_1$.