A total Cuntz semigroup for $C^{\ast }$-algebras of stable rank one

In this paper, we show that for unital, separable $C^{\ast }$-algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor ${\mathrm{K}}_{\ast }$ are naturallly equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable $C^{\ast }$-algebras of stable rank one, is a well-defined continuous functor from the category of $C^{\ast }$-algebras of stable rank one to the category $\underset{‾}{\mathrm{C}\mathrm{u}}$. We prove that this new functor and the functor $\underset{‾}{\mathrm{K}}$ are naturallly equivalent for unital, separable, K-pure $C^{\ast }$-algebras. Therefore, the total Cuntz semigroup is a complete invariant for a large class of $C^{\ast }$-algebras of real rank zero.