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Working paper

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

Zaev D.
We construct an analogue of the classical 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 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 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 p-Wasserstein distance metricizes the weak*-topology on the state space.