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.
Isometries of combinatorial Banach spaces Brech, C.; Ferenczi, V.; Tcaciuc, A.
Proceedings of the American Mathematical Society,
November 1, 2020, 2020-11-00, Letnik:
148, Številka:
11
Journal Article
Recenzirano
Odprti dostop
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of ...Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
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.
We extend existing results that characterize isometries on the Tsirelson-type spaces
T
1
n
,
S
1
T\big \frac {1}{n}, \mathcal {S}_1\big
(
n
∈
N
,
n
⩾
2
n\in \mathbb {N}, n\geqslant 2
) to the ...class
T
θ
,
S
α
T\theta , \mathcal {S}_{\alpha }
(
θ
∈
(
0
,
1
2
\big (\theta \in \big (0, \frac {1}{2}\big
,
1
⩽
α
>
ω
1
1\leqslant \alpha > \omega _1
\big), where
S
α
\mathcal {S}_{\alpha }
denote the Schreier families of order
α
\alpha
. We prove that every isometry on
T
θ
,
S
1
T\theta , \mathcal {S}_1
\big(
θ
∈
(
0
,
1
2
\theta \in \big (0, \frac {1}{2}\big
\big) is determined by a permutation of the first
⌈
θ
−
1
⌉
\lceil {\theta }^{-1} \rceil
elements of the canonical unit basis followed by a possible sign-change of the corresponding coordinates together with a sign-change of the remaining coordinates. Moreover, we show that for the spaces
T
θ
,
S
α
T\theta , \mathcal {S}_{\alpha }
\big(
θ
∈
(
0
,
1
2
\theta \in \big (0, \frac {1}{2}\big
,
2
⩽
α
>
ω
1
2\leqslant \alpha > \omega _1
\big) the isometries exhibit a more rigid character, namely, they are all implemented by a sign-change operation of the vector coordinates.
Abstract
The necessary and sufficient conditions for a spacetime with an invariant frame to admit a group of isometries of dimension
r
are given in terms of the connection tensor
H
associated with ...this frame. In Petrov–Bel types I, II and III, and in other spacetimes where an invariant frame algebraically defined by the curvature tensor exists, the connection tensor
H
is given in terms of the Weyl and Ricci tensors without an explicit determination of the frame. Thus, an intrinsic, deductive, explicit and algorithmic characterization of these spacetimes follows. Some examples show that this algorithm can be easily implemented on the
xAct Mathematica
suite of packages.
The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group of isometries of dimension r acting on s-dimensional orbits are obtained. These conditions are ...Intrinsic, Deductive, Explicit and ALgorithmic and they offer an IDEAL labeling that improves previously known invariant studies.
Let G be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of G with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the ...identity component of the isometry group is compact if G is not simply-connected.
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