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 and isometric imbedding j of F to 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 and each partial isometry of F can be extended up to global isometry in . 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].     

Some embeddings of infinite-dimensional spaces

It is shown that ifA is a weakly infinite-dimensional subset of a metric spaceR then aG _δ setB ofR exists such thatA⊆B andB is weakly infinite-dimensional. A similar result holds for a set having strong transfinite inductive dimension. As a consequence each weakly infinite-dimensional metric space possesses a weakly infinite-dimensional complete metric extension. A similar result holds also for a space having strong transfinite inductive dimension.     

A symmetrized metric ramsey theorem

For every finite metric space A there exists a finite metric space B and a real number r > 0 such that for every coloring of B by two colors there exists a monochromatic A′ ⊆ B such that every isometry between two subsets of A′ extends to a full autoisometry of B and A′ is either isometric to A or is r-homothetic to A.     

On the geometry of Urysohn's universal metric space

In recent years, much interest was devoted to the Urysohn space U and its isometry group; this paper is a contribution to this field of research. We mostly concern ourselves with the properties of isometries of U, showing for instance that any Polish metric space is isometric to the set of fixed points of some isometry φ. We conclude the paper by studying a question of Urysohn, proving that compact homogeneity is the strongest homogeneity property possible in U.     

The isometry group of the Urysohn space as a Lévy group

We prove that the isometry group Iso(U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik stating that Iso(U) has a dense locally finite subgroup.     

Isometries and almost isometries between spaces of continuous functions

We characterize the isometries fromC(X) intoC(Y) whereX andY are compact metric spaces. We give necessary and sufficient conditions on an isometry from a subset ofC(X) intoC(Y) to have an extension to the whole space. It is also shown that an almost isometry from the unit ball ofC(X) into the unit ball ofC(Y) is near to an isometry.     

The Hilbert series of prime PI rings

We study the Hilbert series of finitely generated prime PI algebras. We show that given such an algebraA there exists some finite dimensional subspaceV ofA which contains 1 _A and generatesA as an algebra such that the Hilbert series ofA with respect to the vector spaceV is a rational function.     

Limits of cubes

The celebrated Urysohn space is the completion of a countable universal homogeneous metric space which can itself be built as a direct limit of finite metric spaces. It is our purpose in this paper to give another example of a space constructed in this way, where the finite spaces are scaled cubes. The resulting countable space provides a context for a direct limit of finite symmetric groups with strictly diagonal embeddings, acting naturally on a module which additively is the “Nim field” (the quadratic closure of the field of order 2). Its completion is familiar in another guise: it is the set of Lebesgue-measurable subsets of the unit interval modulo null sets. We describe the isometry groups of these spaces and some interesting subgroups, and give some generalisations and speculations.     

Global optimization in metric spaces with partial orders

The main objective of this article is to resolve an optimization problem in the setting of a metric space that is endowed with a partial order. In fact, given non-empty subsets A and B of a metric space that is equipped with a partial order, and a non-self mapping S: A?→?B, this article explores the existence of an optimal approximate solution, known as a best proximity point of the mapping S, to the equation Sx?=?x, where S is a proximally increasing, ordered proximal contraction. This article exhibits an algorithm for determining such an optimal approximate solution. Moreover, the result elicited in this article subsumes a fixed point theorem, due to Nieto and Rodriguez-Lopez, in the setting of a metric space with a partial order.     

Continuously controlledA-theory

To anycontrolled space over the metric spaceB we can associate itsboundedly controlled algebraic K-theory, a functor designed to give information about the space of stable bounded concordances of manifolds homotopy equivalent toX. Generalizing a construction of D. R. Anderson, F. X. Connolly, S. Ferry, and E. K. Pedersen, we define another functor, calledcontinuously controlled A-theory, which depends only on thetopology of the control space, not itsmetric properties. In the special case whereB=R ₊, this functor is (more or less by definition) the same asproper A-theory. We prove that under certain conditions on the controlled space the natural transformation from boundedly controlledA-theory to continuously controlledA-theory is a weak homotopy equivalence, and hence defines a generalized homology theory. Continuously controlledK-theory is used in the approaches of G. Carlsson, E. K. Pedersen, and S. Ferry, S. Weinbergervto theK-theory Novikov conjecture.     

Algebraic properties of isometries between groups of invertible elements in Banach algebras

We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then (T(eA))⁻¹T is extended to an isometric real algebra isomorphism from A onto B. In particular, A−1 is isometric as a metric space to B−1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of ?elazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.     

Conway’s groupCo 3 and the Dickson invariants

In this paper, we construct a map from the classifying spaceBCo ₃ of Conway’s sporadic simple groupCo ₃ to the classifying spaceBDI(4) of the new finite loop space at the prime two DI(4) of Dwyer and Wilkerson. This map has the property that it injects the mod two cohomology ofBDI(4) (which is equal to the Dickson invariants of rank four) as a subring over which the mod two cohomology ofBCo ₃ is finitely generated as a module.     

Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceB _p and the Dirichlet spaceD ^p . In the case ofB _p we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.     

Almost isometric embedding between metric spaces

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU. We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding. The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ. Research of the first author was supported by an Israeli Science foundation grant no. 177/01. Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827.     

A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph R, the universal Urysohn metric space U, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of U (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that U admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.     

Large superuniversal metric spaces

For every uncountable cardinal κ define a metric spaceS to be κ-superuniversal iff for every metric spaceU of cardinality κ, every partial isometry intoS from a subset ofU of cardinality less than κ can be extended to all ofU. We prove that any such space must have cardinality at least , and for each regular uncountable cardinal κ, we construct a κ-superuniversal metric space of cardinality , It is proved that there is a unique κ-superuniversal metric space of cardinality κ iff . Several decomposition theorems are also proved, e.g., every κ-superuniversal space contains a family of disjoint κ-superuniversal subspaces. Finally, we consider some applications to more general topological spaces, to graph theory, and to category theory, and we conclude with a list of open problems.     

Image d’une somme d’operateurs monotones et applications

LetA andB be monotone (multivalued) operators in a Hilbert spaceH. The paper deals with the relations between the rangeR(A+B) ofA+B and the algebraic sum of the ranges ofA andB, R(A)+R(B).      

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

The Urysohn space embeds in Banach spaces in just one way

In a Master's thesis in 1985 and a subsequent paper published in 1992, the author discovered that the universal separable metric space (up to isometry) discovered by Urysohn in 1925 has a uniquely determined linear closure (up to linear isometry) when isometrically embedded in a Banach space so as to include the zero of the Banach space. The proof of this result is given in this note and the current status of some related questions is discussed.