A quantum metric on the Cantor Space

The first author and Latr\'emoli\`ere had 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.