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

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-similar to. Furthermore, we compute the Cu1-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 circle plus K-1 from Cu-1. (C) 2021 Elsevier Inc. All rights reserved.