A dual formula for the spectral distance in noncommutative geometry

TL;DR

This paper introduces a dual infimum formula for Connes's spectral distance in noncommutative geometry, extending the classical duality in optimal transport to a noncommutative setting, with applications to matrix algebras.

Contribution

It generalizes the dual formula of spectral distance, providing a new perspective and tools for estimating distances in noncommutative geometry.

Findings

Derived a dual infimum formula for spectral distance

Applied the formula to matrix algebras for upper bounds

Extended classical optimal transport duality to noncommutative setting

Abstract

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.