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

TL;DR

This paper proves a homotopy rigidity theorem for certain simple, nuclear, and $ ext{Z}_0$-stable C*-algebras, showing they are isomorphic if trace-preservingly homotopy equivalent, without assuming the UCT.

Contribution

It establishes a new isomorphism criterion for $ ext{Z}_0$-stable C*-algebras based on trace-preserving homotopy equivalence, extending rigidity results.

Findings

Two $ ext{Z}_0$-stable C*-algebras are isomorphic if trace-preservingly homotopy equivalent.

The result applies without the UCT assumption.

Analogous to the homotopy rigidity for Kirchberg algebras.

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.