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Nuclear dimension of extensions of commutative C*-algebras by Kirchberg algebras

Samuel Evington, Abraham C. S. Ng, Aidan Sims, Stuart White

TL;DR

This work determines the nuclear dimension of essential extensions 0→J→E→C(X)→0 where J is a stable Kirchberg algebra and the quotient is commutative. The authors develop and deploy a Brake–Winter decomposition, extend order-zero maps via $\mathcal{O}_\infty$-stable uniqueness (via Gabe) and absorption, and implement a two-colour approximation scheme to handle the $\dim X=1$ case, culminating in a sharp bound. They prove that for such extensions $\mathrm{dim}_\mathrm{nuc}(E)=\max(1,\dim X)$, with $\dim X$ finite; the result aligns with known cases and extends the Brake–Winter framework to purely infinite ideals. The graph-algebra perspective is explored, yielding a geometric criterion and illustrating the theorem’s reach to finite graph C*-algebras and their ideal lattices, thereby informing broader regularity conjectures in the Elliott program.

Abstract

We compute the nuclear dimension of extensions of C*-algebras involving commutative unital quotients and stable Kirchberg ideals. We identify the finite directed graphs whose C*-algebras are covered by this theorem.

Nuclear dimension of extensions of commutative C*-algebras by Kirchberg algebras

TL;DR

This work determines the nuclear dimension of essential extensions 0→J→E→C(X)→0 where J is a stable Kirchberg algebra and the quotient is commutative. The authors develop and deploy a Brake–Winter decomposition, extend order-zero maps via -stable uniqueness (via Gabe) and absorption, and implement a two-colour approximation scheme to handle the case, culminating in a sharp bound. They prove that for such extensions , with finite; the result aligns with known cases and extends the Brake–Winter framework to purely infinite ideals. The graph-algebra perspective is explored, yielding a geometric criterion and illustrating the theorem’s reach to finite graph C*-algebras and their ideal lattices, thereby informing broader regularity conjectures in the Elliott program.

Abstract

We compute the nuclear dimension of extensions of C*-algebras involving commutative unital quotients and stable Kirchberg ideals. We identify the finite directed graphs whose C*-algebras are covered by this theorem.
Paper Structure (8 sections, 16 theorems, 63 equations)

This paper contains 8 sections, 16 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Machinery of proof
  4. Extending order zero maps
  5. Ensuring injectivity
  6. Approximating the Kirchberg ideal
  7. Proof of Theorem \ref{['thm:main']}
  8. Graph algebras

Key Result

Theorem B

Let $J$ be a stable Kirchberg algebra and $X$ a compact metric space. Let be an essential extension. Then $\mathrm{dim}_\mathrm{nuc}(E) = \max(1,\dim(X))$.

Theorems & Definitions (34)

  • Theorem B
  • Definition 1: c.f. TW
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Proposition 1: The Brake--Winter decomposition
  • proof
  • Proposition 2
  • proof
  • ...and 24 more