Inductive limits of c*-algebras and compact quantum metric spaces

Konrad Aguilar*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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 languageEnglish
JournalJournal of the Australian Mathematical Society
Volume111
Issue number3
Pages (from-to)289-312
ISSN1446-7887
DOIs
Publication statusPublished - 27. Dec 2021

Keywords

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

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