### Abstract

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

Pages (from-to) | 4631-4670 |

Number of pages | 40 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 7 |

Early online date | 27 Dec 2016 |

DOIs | |

Publication status | Published - Jul 2017 |

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*Transactions of the American Mathematical Society*,

*369*(7), 4631-4670. https://doi.org/10.1090/tran/6842

**Nuclear dimension and Z-stability of non-simple C*-algebras.** / Robert, Leonel; Tikuisis, Aaron.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 369, no. 7, pp. 4631-4670. https://doi.org/10.1090/tran/6842

}

TY - JOUR

T1 - Nuclear dimension and Z-stability of non-simple C*-algebras

AU - Robert, Leonel

AU - Tikuisis, Aaron

PY - 2017/7

Y1 - 2017/7

N2 - We investigate the interplay of the following regularity properties for non-simple C ∗ -algebras: finite nuclear dimension, Z-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic regularity in the Cuntz semigroup, provided that known type I obstructions are avoided. We demonstrate how finite nuclear dimension can be used to study the structure of the central sequence algebra, by factorizing the identity map on the central sequence algebra, in a manner resembling the factorization arising in the definition of nuclear dimension. Results about the central sequence algebra are used to attack the conjecture that finite nuclear dimension implies Z-stability, for sufficiently non-type I, separable C∗ -algebras. We prove this conjecture in the following cases: (i) the C∗ -algebra has no simple purely infinite ideals of quotients and its primitive ideal space has a basis of compact open sets, (ii) the C ∗ -algebra has no purely infinite quotients and its primitive ideal space is Hausdorff. In particular, this covers C∗ -algebras with finite decomposition rank and real rank zero. Our results hold more generally for C∗ -algebras with locally finite nuclear dimension which are (M, N)-pure (a regularity condition of the Cuntz semigroup).

AB - We investigate the interplay of the following regularity properties for non-simple C ∗ -algebras: finite nuclear dimension, Z-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic regularity in the Cuntz semigroup, provided that known type I obstructions are avoided. We demonstrate how finite nuclear dimension can be used to study the structure of the central sequence algebra, by factorizing the identity map on the central sequence algebra, in a manner resembling the factorization arising in the definition of nuclear dimension. Results about the central sequence algebra are used to attack the conjecture that finite nuclear dimension implies Z-stability, for sufficiently non-type I, separable C∗ -algebras. We prove this conjecture in the following cases: (i) the C∗ -algebra has no simple purely infinite ideals of quotients and its primitive ideal space has a basis of compact open sets, (ii) the C ∗ -algebra has no purely infinite quotients and its primitive ideal space is Hausdorff. In particular, this covers C∗ -algebras with finite decomposition rank and real rank zero. Our results hold more generally for C∗ -algebras with locally finite nuclear dimension which are (M, N)-pure (a regularity condition of the Cuntz semigroup).

U2 - 10.1090/tran/6842

DO - 10.1090/tran/6842

M3 - Article

VL - 369

SP - 4631

EP - 4670

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 7

ER -