## Abstract

Given a unital inductive limit of C∗-algebras for which each C∗-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov-Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov-Hausdorff propinquity topology.

Original language | English |
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Journal | Journal of the Australian Mathematical Society |

Volume | 111 |

Issue number | 3 |

Pages (from-to) | 289-312 |

ISSN | 1446-7887 |

DOIs | |

Publication status | Published - 27. Dec 2021 |

## Keywords

- AF algebras
- inductive limits
- Lip-norms
- Monge-Kantorovich distance
- noncommutative metric geometry
- quantum metric spaces