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A quantum
metric on the Cantor space
Konrad Aguilar and Alejandra López |
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Vol. 16 (2023), No. 5, 737–764
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Abstract |
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Aguilar and Latrémolière introduced a quantum metric (in the sense of Rieffel) on the algebra of complex-valued continuous functions on the Cantor space. We show that this quantum metric is distinct from the quantum metric induced by a classical metric on the Cantor space. We accomplish this by showing that the seminorms induced by each quantum metric (Lip-norms) are distinct on a dense subalgebra of the algebra of complex-valued continuous functions on the Cantor space. In the process, we develop formulas for each Lip-norm on this dense subalgebra and show these Lip-norms agree on a Hamel basis of this subalgebra. Then, we use these formulas to find families of elements for which these Lip-norms disagree. |
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Keywords
noncommutative metric geometry, quantum metric spaces,
Lip-norms, C*-algebras, Cantor space
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Mathematical Subject Classification 2010
Primary: 46L30, 46L89, 58B34
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Milestones
Received: 16 March 2020
Revised: 28 August 2022
Accepted: 12 November 2022
Published: 9 December 2023
Communicated by Kenneth S. Berenhaut
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Authors |
| Konrad Aguilar | |
| Department of Mathematics and
Statistics Pomona College Claremont, CA United States |
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| Alejandra López | |
| Department of Mathematics Purdue University West Lafayette, IN United States |
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| © 2023 MSP (Mathematical Sciences Publishers). |
