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

TL;DR

This paper introduces the total Cuntz semigroup, a new invariant for certain $C^*$-algebras of stable rank one, and proves its equivalence to $K$-theory, enhancing classification tools.

Contribution

It defines the total Cuntz semigroup as a refined invariant and establishes its natural equivalence to $K$-theory for a broad class of $C^*$-algebras of stable rank one.

Findings

Total Cuntz semigroup is a well-defined continuous functor.

It is naturally equivalent to $K$-theory for unital, separable, K-pure $C^*$-algebras.

The invariant is complete for a large class of real rank zero $C^*$-algebras.

Abstract

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 ${\rm K}_*$ 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 $C^*$-algebras of stable rank one, is a well-defined continuous functor from the category of $C^*$-algebras of stable rank one to the category ${\rm\underline{ Cu}}$. We prove that this new functor and the functor ${\rm \underline{K}}$ are naturallly 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.