### Abstract

Original language | English |
---|---|

Pages (from-to) | 729-778 |

Number of pages | 42 |

Journal | Mathematische Annalen |

Volume | 358 |

Issue number | 3-4 |

Early online date | 21 Sep 2013 |

DOIs | |

Publication status | Published - Apr 2014 |

### Fingerprint

### Keywords

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

### Cite this

*Mathematische Annalen*,

*358*(3-4), 729-778. https://doi.org/10.1007/s00208-013-0951-0

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

Research output: Contribution to journal › Article

*Mathematische Annalen*, vol. 358, no. 3-4, pp. 729-778. https://doi.org/10.1007/s00208-013-0951-0

}

TY - JOUR

T1 - Nuclear dimension, Z -stability, and algebraic simplicity for stably projectionless C∗ -algebras

AU - Tikuisis, Aaron

N1 - Fixed typos, etc., and added corollary regarding finitely many ideals. 37 pages. To appear in Mathematische Annalen. The published version will differ

PY - 2014/4

Y1 - 2014/4

N2 - 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.

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.

KW - math.OA

KW - math.FA

KW - 46L35, 46L80, 46L05, 47L40, 46L85

U2 - 10.1007/s00208-013-0951-0

DO - 10.1007/s00208-013-0951-0

M3 - Article

VL - 358

SP - 729

EP - 778

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 3-4

ER -