Non-Commutative Metrics on Matrix State Spaces

We use the theory of quantization to introduce matrix versions of metric on the state space of a C ∗-algebra and Lipschitz seminorm on the algebra. We show that a lower semicontinuous matrix Lipschitz seminorm is determined by its associated 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. These results generalize results of Rieffel to the quantized situation.