On quantum symmetries of compact metric spaces

An action of a compact quantum group on a compact metric space (X,d)(X,d) is (D)-isometric if the distance function is preserved by a diagonal action on X×XX×X. In this study, we show that an isometric action in this sense has the following additional property: the corresponding action on the algebra of continuous functions on XX by the convolution semigroup of probability measures on the quantum group contracts Lipschitz constants. In other words, it is isometric in another sense due to Li, Quaegebeur, and Sabbe, which partially answers a question posed by Goswami. We also introduce other possible notions of isometric quantum actions in terms of the Wasserstein pp-distances between probability measures on XX for p≥1p≥1, which are used extensively in optimal transportation. Indeed, all of these definitions of quantum isometry belong to a hierarchy of implications, where the two described above lie at the extreme ends of the hierarchy. We conjecture that they are all equivalent.