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

2015/05/22 by Zaev, Danila
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1505.06061

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 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 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.

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