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
