Isometries Of Metric Spaces Research Articles

The Groups of Isometries of Metric Spaces over Vector Groups

In this paper, we consider the groups of isometries of metric spaces arising from finitely generated additive abelian groups. Let A be a finitely generated additive abelian group. Let R={1,ϱ} where ϱ is a reflection at the origin and T={ta:A→A,ta(x)=x+a,a∈A}. We show that (1) for any finitely generated additive abelian group A and finite generating set S with 0∉S and −S=S, the maximum subgroup of IsomX(A,S) is RT; (2) D⊴RT if and only if D≤T or D=RT′ where T′={h2:h∈T}; (3) for the vector groups over integers with finite generating set S={u∈Zn:|u|=1}, IsomX(Zn,S)=On(Z)Zn. The paper also includes a few intermediate technical results.

AUTOMATIC CONTINUITY FOR ISOMETRY GROUPS

We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\text{Aut}([0,1],\unicode[STIX]{x1D706})$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.

Finitely approximate groups and actions Part I: The Ribes–Zalesskiĭ property

AbstractWe investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces [11] and obtain the following exact equivalence: any action of a discrete group Γ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of Γ is closed in the profinite topology on Γ.

Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U , namely Theorems A and G: Theorem A (Approximation): The group of isometry ISO ( U ) contains everywhere dense locally finite subgroup; Theorem G (Globalization): For each finite metric space F there exists another finite metric space F ¯ and isometric imbedding j of F to F ¯ such that isometry j induces the imbedding of the group monomorphism of the group of isometries of the space F to the group of isometries of space F ¯ and each partial isometry of F can be extended up to global isometry in F ¯ . The fact that Theorem G, is true was announced in 2005 by author without proof, and was proved by S. Solecki in [S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005) 315–332] (see also [V. Pestov, The isometry group of the Urysohn space as a Lévy group, Topology Appl. 154 (10) (2007) 2173–2184; V. Pestov, A theorem of Hrushevski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space, math/0702207]) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from [P. Cameron, A. Vershik, Some isometry groups of Urysohn spaces, Ann. Pure Appl. Logic 143 (1–3) (2006) 70–78].

Error-block codes and poset metrics

Let $P = (${$1, 2,\ldots,n$}$,$≤$)$ be a poset, let $V_1, V_2,\ldots, V_n$ be a family of finite-dimensional spaces over a finite field $\mathbb F_q$ and let <br> $ V = V_1 \oplus V_2 \oplus\ldots \oplus V_n.$ <br> In this paper we endow $V$ with a poset metric such that the $P$-weight is constant on the non-null vectors of a component $V_i$, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code $[8; 4; 4]$ to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code $[24; 12; 8]$ into perfect codes. We also give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometries of the error-block metric spaces.

Homotheties and isometries of metric spaces

Homotheties and isometries of metric spaces

Isometries of quantum states

This paper treats the isometries of metric spaces of quantum states. We consider two metrics on the set all quantum states, namely the Bures metric and the one which comes from the trace-norm. We describe all the corresponding (nonlinear) isometries and also present similar results concerning the space of all (non-normalized) density operators.

Isometries of spaces of unbounded continuous functions

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

A generalization of isometries to uniform spaces

Isomorphisms are, in many ways, the generalizations of isometrics to uniform spaces. Yet some theorems on isometries of metric spaces only generalize to uniform spaces in terms of more restricted transformations of the uniform space. In § 1, in the course of a discussion of a theorem on transitive groups of automorphisms, we define such a transformation and call it anisobasism. It appears that in many respects isobasisms, rather than isomorphisms, are the generalizations of isometries to uniform spaces. The results of Freudenthal and Hurewicz (7) on contractions, expansions and isometries of totally bounded metric spaces are generalized, in § 2, to contractions, expansions and isobasisms of totally bounded uniform spaces. These results, together with generalizations of some theorems of Eilenberg (6) on compact groups of homeomorphisms of metric spaces which are obtained in §3, give a characterization of isobasisms. The language of Bourbaki (2,3,4) is used throughout this note.