The nuclear dimension of $\mathcal O_\infty$-stable $C^*$-algebras

TL;DR

This paper proves that nuclear $ ext{O}_ ext{infty}$-stable C*-algebras and homomorphisms have nuclear dimension at most 1, providing a classification criterion based on quasidiagonality and primitive-ideal structure.

Contribution

It establishes the nuclear dimension bounds for $ ext{O}_ extinfty$-stable C*-algebras and characterizes conditions for finite decomposition rank.

Findings

Nuclear $ ext{O}_ extinfty$-stable C*-algebras have nuclear dimension 1.

Homomorphisms with certain stability have nuclear dimension at most 1.

Criteria for finite decomposition rank involve quasidiagonality and primitive ideals.

Abstract

We show that every nuclear $\mathcal O_\infty$-stable *-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, $\mathcal O_\infty$-stable C*-algebras have nuclear dimension 1. We also characterise when $\mathcal O_\infty$-stable C*-algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full $\mathcal O_2$-stable *-homomorphisms have nuclear dimension 0.