Nuclear dimension, Z -stability, and algebraic simplicity for stably projectionless C∗ -algebras

Aaron Tikuisis

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [Toms, "K-theoretic rigidity and slow dimension growth"; Winter, "Nuclear dimension and Z-stability of pure C*-algebras"] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C*-algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept.
Original languageEnglish
Pages (from-to)729-778
Number of pages42
JournalMathematische Annalen
Volume358
Issue number3-4
Early online date21 Sep 2013
DOIs
Publication statusPublished - Apr 2014

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C*-algebra
Simplicity
Unital
Rigidity
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Imply
Algebra
Generalise

Keywords

  • math.OA
  • math.FA
  • 46L35, 46L80, 46L05, 47L40, 46L85

Cite this

Nuclear dimension, Z -stability, and algebraic simplicity for stably projectionless C∗ -algebras. / Tikuisis, Aaron.

In: Mathematische Annalen, Vol. 358, No. 3-4, 04.2014, p. 729-778.

Research output: Contribution to journalArticle

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AB - The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [Toms, "K-theoretic rigidity and slow dimension growth"; Winter, "Nuclear dimension and Z-stability of pure C*-algebras"] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C*-algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept.

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