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

In this paper, we show that for unital, separable C* -algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor K* are natural equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable C*-algebras of stable rank one, is a well-defined continuous functor from the category of C*- algebras of stable rank one to the category Cu. We prove that this new functor and the functor K are naturally equivalent for unital, separable, K-pure C*-algebras. Therefore, the total Cuntz semigroup is a complete invariant for a large class of C*-algebras of real rank zero.(c) 2023 Elsevier Inc. All rights reserved.