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A quantum metric on the Cantor space

Konrad Aguilar and Alejandra López

Vol. 16 (2023), No. 5, 737–764
Abstract

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.

Keywords
noncommutative metric geometry, quantum metric spaces, Lip-norms, C*-algebras, Cantor space
Mathematical Subject Classification 2010
Primary: 46L30, 46L89, 58B34
Milestones
Received: 16 March 2020
Revised: 28 August 2022
Accepted: 12 November 2022
Published: 9 December 2023

Communicated by Kenneth S. Berenhaut
Authors
Konrad Aguilar
Department of Mathematics and Statistics
Pomona College
Claremont, CA
United States
Alejandra López
Department of Mathematics
Purdue University
West Lafayette, IN
United States