$L^p$-Wasserstein distances on state and quasi-state spaces of $C^{\ast }$-algebras

Abstract

We construct an analogue of the classical $L^p$-Wasserstein distance for the state space of a $C^{\ast }$-algebra. Given an abstract Lipschitz gauge on a $C^{\ast }$-algebra $\mathcal{A}$ in the sense of Rieffel, one can define the classical $L^p$-Wasserstein distance on the state space of each commutative $C^{\ast }$-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 $C^{\ast }$-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${}^{\ast }$-topology on the state space.

Preprint version. V.2 major changes: serious mistake in Prop. 4.11 corrected, changes in Prop. 5.11, Th. 5.14, some small improvements

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