The Fell topology and the modular Gromov-Hausdorff propinquity
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- by Konrad Aguilar and Jiahui Yu
- Proc. Amer. Math. Soc. 152 (2024), 1711-1724
- DOI: https://doi.org/10.1090/proc/16669
- Published electronically: February 15, 2024
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Abstract:
We show that any (norm-closed two-sided) ideal of a unital C*-algebra that comes equipped with certain natural quantum metric structure is a metrized quantum vector bundle, when canonically viewed as a module over $A$. Next, given a unital AF-algebra $A$ equipped with a faithful tracial state, we equip each ideal of $A$ with a metrized quantum vector bundle structure using previous work of the first author and Latrémolière. Moreover, we show that convergence of ideals in the Fell topology implies convergence of the associated metrized quantum vector bundles in the modular Gromov-Hausdorff propinquity of Latrémolière. In a similar vein but requiring a different approach, given a compact metric space $(X,d)$, we equip each ideal of $C(X)$ with a metrized quantum vector bundle structure, and show that convergence in the Fell topology implies convergence in the modular Gromov-Hausdorff propinquity.References
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Bibliographic Information
- Konrad Aguilar
- Affiliation: Department of Mathematics and Statistics, Pomona College, 610 N. College Ave., Claremont, California 91711
- MR Author ID: 1150712
- ORCID: 0000-0003-2766-4694
- Email: konrad.aguilar@pomona.edu
- Jiahui Yu
- Affiliation: Department of Mathematics and Statistics, Pomona College, 610 N. College Ave., Claremont, California 91711
- MR Author ID: 1410234
- Email: jyad2018@mymail.pomona.edu
- Received by editor(s): November 20, 2022
- Received by editor(s) in revised form: June 20, 2023, and September 11, 2023
- Published electronically: February 15, 2024
- Additional Notes: The first author was financially supported by the Independent Research Fund Denmark through the project “Classical and Quantum Distances” (grant no. 0940-00107B).
- Communicated by: Adrian Ioana
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 1711-1724
- MSC (2020): Primary 46L89, 46L30, 58B34
- DOI: https://doi.org/10.1090/proc/16669
- MathSciNet review: 4709237