A dual formula for the spectral distance in noncommutative geometry

In noncommutative geometry, Connes's spectral distance is an extended metric on the state space of a C∗-algebra generalizing Kantorovich's dual formula of the Wasserstein distance of order 1 from optimal transport. It is expressed as a supremum. We present a dual formula – as an infimum – generalizing Beckmann's “dual of the dual” formulation of the Wasserstein distance. We then discuss some examples with matrix algebras, where such a dual formula may be useful to obtain upper bounds for the distance.