Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero

Qingnan An; Zhichao Liu

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arXiv · 2303.06835

Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero

Qingnan An·Zhichao Liu

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Abstract

In this paper, we exhibit two unital, separable, nuclear ${\rm C}^*$-algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of total K-theory. For the general setting, with a new invariant -- total Cuntz semigroup \cite{AL}, we classify a large class of ${\rm C}^*$-algebras obtained from extensions. The total Cuntz semigroup, which distinguish the algebras of our counterexample, could possibly classify all the ${\rm C}^*$-algebras of stable rank one and real rank zero.

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BibTeX RIS

Qingnan An, Zhichao Liu. 2023-03-13. Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero. https://arxiv.org/abs/2303.06835

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