A homotopy rigidity theorem for $\mathcal{Z}_0$-stable $\mathrm{C}^\ast$-algebras

TL;DR

This work addresses rigidity for simple, separable, nuclear $\mathcal{Z}_0$-stable $C^*$-algebras without assuming the UCT. It develops a ${\mathcal{Z}_0}$-stable uniqueness principle for maps that are trace-preservingly homotopic, leveraging the augmented Cuntz semigroup and Robert's classification framework. The authors prove that trace-preserving homotopy equivalence between two such algebras forces an isomorphism after tensoring with $\mathcal{Z}_0$, via an Elliott intertwining argument, thereby providing a stably projectionless analogue of homotopy rigidity results for Kirchberg algebras. This yields a non-UCT rigidity result and aligns with, yet extends, Schafhauser’s 2024 theorem by focusing on the stably projectionless setting and requiring a trace-preserving homotopy in both directions. The findings contribute to a broader understanding of classification and rigidity outside the UCT paradigm, particularly in the stably finite regime.

Abstract

We show that two simple, separable, nuclear and $\mathcal{Z}_0$-stable $\mathrm{C}^\ast$-algebras are isomorphic if they are trace-preservingly homotopy equivalent. This result does not assume the UCT and can be viewed as a tracial stably projectionless analog of the homotopy rigidity theorem for Kirchberg algebras.

Theorems & Definitions (19)

• Theorem A
• Theorem A: Rob12
• Theorem B: Sza21, Elliott2020
• Theorem C: cf. GL20
• Theorem D: cf. GL20
• Lemma E: GL20
• Lemma A
• proof
• Lemma B
• proof