Let (X,d,m) be a compact non-branching metric measure space equipped with a qualitatively non-degenerate measure m. The study of properties of the Lp–Wasserstein space (Pp(X),Wp) associated to X has ...proved useful in describing several geometrical properties of X. In this paper we focus on the study of isometries of Pp(X) for p∈(1,∞) under the assumption that there is some characterization of optimal maps between measures, the so called Good Transport Behaviour GTBp. Our first result states that the set of Dirac deltas is invariant under isometries of the Lp–Wasserstein space. Additionally, for Riemannian manifolds we obtain that the isometry groups of the Lp–Wasserstein space and of the base space coincide under geometric assumptions on the manifold; namely, for p=2 that the sectional curvature is strictly positive and for general p∈(1,∞) that the Riemannian manifold is a Compact Rank One Symmetric Space.
Suppose that a compact quantum group Q acts faithfully on a smooth, compact, connected manifold M, i.e. has a C⁎ (co)-action α on C(M), such that α(C∞(M))⊆C∞(M,Q) and the linear span of α(C∞(M))(1⊗Q) ...is dense in C∞(M,Q) with respect to the Fréchet topology. It was conjectured by the author quite a few years ago that Q must be commutative as a C⁎ algebra i.e. Q≅C(G) for some compact group G acting smoothly on M. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.
The MacWilliams' Extension Theorem (MET) with respect to a combinatorial metric states that every isomorphism between linear codes that preserves combinatorial weight can be extended to an isometric ...extension of the whole space. Pinheiro et al. (2019) proposed the problem of characterizing combinatorial metrics over a finite field with two elements that admit the MET. In this paper, we provide the complete description of such metrics.
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