Abstract

We construct Cartan subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan subalgebra.

Highlights

  • Classification of C*-algebras has seen tremendous advances recently

  • The classification of unital separable simple nuclear Z-stable C*algebras satisfying the UCT is complete. This is the culmination of work by many mathematicians

  • I would like to thank the referee for pointing out that [11] describes a general construction exhausting all possible Elliott invariants with weakly unperforated pairing between K-theory and traces

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Summary

Introduction

Classification of C*-algebras has seen tremendous advances recently. In the unital case, the classification of unital separable simple nuclear Z-stable C*algebras satisfying the UCT is complete. It is expected that—once the stably projectionless case is settled—the final result will classify all separable simple nuclear Z-stable C*-algebras satisfying the UCT by their Elliott invariants. This class of C*-algebras is what we refer to as “classifiable C*-algebras”. To complete these classification results, it is important to construct concrete models realizing all possible Elliott invariants by classifiable C*-algebras. The key tool for all the results in this paper is an improved version of [4, Theorem 3.6], which allows us to construct Cartan subalgebras in inductive limit C*-algebras. I would like to thank the referee for very helpful comments which led to an improved version of Theorem 1.3 (previous versions of this theorem only covered classifiable stably projectionless C*-algebras with trivial pairing between K-theory and traces)

The unital case
The stably projectionless case
Concrete construction of AH-algebras
The general construction with concrete models
Summary of the construction
Inductive limits and Cartan pairs revisited
The building blocks
The connecting maps
Conclusions
Examples

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