  We use the theory of quantization to introduce non-commutative versions of
metric on state space and Lipschitz seminorm. We show that a lower
semicontinuous matrix Lipschitz seminorm is determined by their matrix metrics
on the matrix state spaces. A matrix metric comes from a lower semicontinuous
matrix Lip-norm if and only if it is convex, midpoint balanced, and midpoint
concave. The operator space of Lipschitz functions with a matrix norm coming
from a closed matrix Lip-norm is the operator space dual of an operator space.
They generalize Rieffel's results to the quantized situation.
