$L^p$-Wasserstein distances on state and quasi-state spaces of $C^*$-algebras

TL;DR

This paper develops a new $L^p$-Wasserstein distance for the state space of $C^*$-algebras, extending classical concepts to noncommutative settings and analyzing its topological properties.

Contribution

It introduces a projective $L^p$-Wasserstein distance on $C^*$-algebra state spaces, generalizing classical Wasserstein distances to noncommutative frameworks.

Findings

The distance is well-behaved and compatible with the topology of the state space.

It provides a sufficient condition for the distance to metrize the weak$^*$-topology.

The construction extends classical Wasserstein distances to quasi-linear states.

Abstract

We construct an analogue of the classical $L^p$-Wasserstein distance for the state space of a $C^*$-algebra. Given an abstract Lipschitz gauge on a $C^*$-algebra $\mathcal{A}$ in the sense of Rieffel, one can define the classical $L^p$-Wasserstein distance on the state space of each commutative $C^*$-subalgebra of $\mathcal{A}$. We consider a projective limit of these metric spaces, which appears to be the space of all quasi-linear states, equipped with a distance function. We call this distance the projective $L^p$-Wasserstein distance. It is easy to show, that the state space of a $C^*$-algebra is naturally embedded in the space of its quasi-linear states, hence, the introduced distance is defined on the state space as well. We show that this distance is reasonable and well-behaved. We also formulate a sufficient condition for a Lipschitz gauge, such that the corresponding projective $L^p$-Wasserstein distance metricizes the weak$^*$-topology on the state space.