Lipschitz isometries of compact quantum metric spaces

A compact quantum metric space is a complete order unit space endowed with a Lip-norm. We investigate the category morphism between compact quantum metric spaces. We show that a unital positive linear map between compact quantum metric spaces with lower semicontinuous Lip-norms is Lipschitz if and only if it preserves the Lipschitz elements. We characterize when a unital positive linear map on a compact quantum metric space with lower semicontinuous Lip-norm is a Lipschitz isometry.