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Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras

José R. Carrión, James Gabe, Christopher Schafhauser, Aaron Tikuisis, Stuart White

TL;DR

The paper develops an abstract, embedding-centric approach to classifying unital embeddings of a broad class of C*-algebras into $\mathcal{Z}$-stable, unital simple nuclear algebras in the UCT class. Central to the method is the total invariant $\underline{K}T_u$, augmented with trace data and the Hausdorffized algebraic $K_1$, together with a trace-kernel extension that splits the problem into tractable trace-based and lift-based steps. The authors establish KK- and KL-uniqueness results in the $\mathcal{Z}$-stable setting, prove existence and classification of lifts, and compute the relevant $KL(A,J_B)$ via the universal (multi)coefficient theorem, culminating in a unital classification theorem that recovers Elliott’s invariant as the classifier in the stably finite case. The work provides a conceptually streamlined, largely self-contained route to the unital classification, with broad implications for automatic structure results and range of invariant realizations in the class of classifiable algebras.

Abstract

We classify the unital embeddings of a unital separable nuclear $C^*$-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^*$-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^*$-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$-theory and traces.

Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras

TL;DR

The paper develops an abstract, embedding-centric approach to classifying unital embeddings of a broad class of C*-algebras into -stable, unital simple nuclear algebras in the UCT class. Central to the method is the total invariant , augmented with trace data and the Hausdorffized algebraic , together with a trace-kernel extension that splits the problem into tractable trace-based and lift-based steps. The authors establish KK- and KL-uniqueness results in the -stable setting, prove existence and classification of lifts, and compute the relevant via the universal (multi)coefficient theorem, culminating in a unital classification theorem that recovers Elliott’s invariant as the classifier in the stably finite case. The work provides a conceptually streamlined, largely self-contained route to the unital classification, with broad implications for automatic structure results and range of invariant realizations in the class of classifiable algebras.

Abstract

We classify the unital embeddings of a unital separable nuclear -algebra satisfying the universal coefficient theorem into a unital simple separable nuclear -algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear -algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of -theory and traces.
Paper Structure (72 sections, 88 theorems, 258 equations)

This paper contains 72 sections, 88 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Context and examples
  3. $\mathcal{Z}$-stability
  4. The UCT
  5. The invariant
  6. Examples
  7. A brief history of the unital classification theorem
  8. Early days
  9. Purely infinite $C^*$-algebras
  10. Approximately type I $C^*$-algebras
  11. Tracially AF $C^*$-algebras
  12. Localization and rational tracial approximation
  13. The Gong--Lin--Niu class and classification by embeddings
  14. Stable uniqueness across the interval and quasidiagonality
  15. Post-2015: $\mathcal{Z}$-stability and abstract classification
  16. ...and 57 more sections

Key Result

Theorem A

Unital simple separable nuclear $\mathcal{Z}$-stable $C^*$-algebras satisfying Rosenberg and Schochet's universal coefficient theorem are classified by Elliott's invariant consisting of $K$-theory and traces.

Theorems & Definitions (205)

  • Theorem A: The unital classification theorem; see Theorem \ref{['algebra-classification']}
  • Theorem B: Classification of unital embeddings; see Theorem \ref{['one-sided-classification']}
  • Corollary C
  • Theorem A: Classification of unital full approximate embeddings; see Theorem \ref{['approximate-classification']}
  • Theorem B: Classification of unital lifts along the trace-kernel extension; see Theorem \ref{['thm:ClassifyingUnitalLifts']}
  • Theorem C: Classification of unital morphisms into the trace-kernel quotient; see Theorem \ref{['thm:B^inftyClassification']}
  • Theorem D: $\mathcal{Z}$-stable $KK$-uniqueness theorem; see Theorem \ref{['thm:KK-Uniqueness']}\ref{['KK-Uniqueness']}
  • Theorem E: Calculating $K_1(J_B)$; see Theorem \ref{['Thm:K1J']}
  • Theorem F: Calculating $KL(A, J_B)$; see Theorem \ref{['thm:calcKL']}
  • Proposition A
  • ...and 195 more