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INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

Part of: Selfadjoint operator algebras Infinite-dimensional manifolds

Published online by Cambridge University Press:  27 March 2020

KONRAD AGUILAR
KONRAD AGUILAR*
Affiliation:
Department of Mathematics and Computer Science (IMADA), University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark e-mail: aguilar@imada.sdu.dk
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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.

Keywords

noncommutative metric geometryMonge–Kantorovich distancequantum metric spacesLip-normsinductive limitsAF algebras

MSC classification

Primary: 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory 46L30: States 58B34: Noncommutative geometry (à la Connes)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 289 - 312
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by L. O. Clark

We gratefully acknowledge the financial support from the Independent Research Fund Denmark through the ‘Classical and Quantum Distances’ project (Grant No. 9040-00107B).

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