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

Qingnan An; Zhichao Liu

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arXiv · 2201.10763

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

Qingnan An·Zhichao Liu

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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.

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BibTeX RIS

Qingnan An, Zhichao Liu. 2022-01-26. A total Cuntz semigroup for $C^*$-algebras of stable rank one. https://arxiv.org/abs/2201.10763

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