arXiv subjects

Vladimir G. Pestov

Publications and source records attributed to Vladimir G. Pestov.

32 records · Page 2Linked to original sources

arXiv2011

Sur les espaces test pour la moyennabilité

We observe that a Polish group $G$ is amenable if and only if every continuous action of $G$ on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the Cantor space can be used to detect amenability and extreme amenability of Polish non-archimedean groups as well as amenability at infinity of discrete countable groups. As corollary, the latter property can also be tested by actions on the Hilbert cube. These results generalize a criterion due to Giordano and de la Harpe.

math.GR↗

arXiv2011

Subgroups of isometries of Urysohn-Katetov metric spaces of uncountable density

According to Kat\vetov (1988), for every infinite cardinal $\mathfrak m$ satisfying ${\mathfrak m}^{\mathfrak n}\leq {\mathfrak m}$ for all ${\mathfrak n}<{\mathfrak m}$, there exists a unique $\mathfrak m$-homogeneous universal metric space $\Ur_{\mathfrak m}$ of weight $\mathfrak m$. This object generalizes the classical Urysohn universal metric space $\Ur = \Ur_{\aleph_0}$. We show that for $\mathfrak m$ uncountable, the isometry group $\Iso(\Urm)$ with the topology of simple convergence is not a universal group of weight $\mathfrak m$: for instance, it does not contain $\Iso(\Ur)$ as a topological subgroup. More generally, every topological subgroup of $\Iso(\Urm)$ having density $<{\mathfrak m}$ and possessing the bounded orbit property $(OB)$ is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskij's 1990 result about the group $\Iso(\Ur)$ being a universal Polish group.

math.GN↗

arXiv2008

Fixed point-free isometric actions of topological groups on Banach spaces

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope of the universal Urysohn metric space, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group of the separable infinite dimensional Hilbert space with strong operator topology, the infinite symmetric group, etc.

math.GR↗

arXiv2008

Hyperlinear and sofic groups: a brief guide

Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless. Hyperlinear groups have their origin in Connes' Embedding Conjecture about von Neumann factors of type $II_1$, while sofic groups, introduced by Gromov, are motivated by Gottschalk Surjunctivity Conjecture (can a shift $A^G$ contain a proper isomorphic copy of itself, where $A$ is a finite discrete space and $G$ is a group?). Groups from both classes can be characterized as subgroups of metric ultraproducts of families of certain metric groups (formed in the same way as ultraproducts of Banach spaces): unitary groups of finite rank lead to hyperlinear groups, symmetric groups of finite rank - to sofic groups. We offer an introductory guide to some of the main concepts, results, and sources of the theory, following Connes, Gromov, Benjamin Weiss, Kirchberg, Ozawa, Radulescu, Elek and Szabó, and others, and discuss open questions which are for the time being perhaps more numerous than the results.

math.GR↗

arXiv2007

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 $\Ur$, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of $\Ur$ (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 $\Ur$ admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.

math.MG↗

arXiv2006

Representations of residually finite groups by isometries of the Urysohn space

As a consequence of Kirchberg's work, Connes' Embedding Conjecture is equivalent to the property that every homomorphism of the group $F_\infty\times F_\infty$ into the unitary group $U(\ell^2)$ with the strong topology is pointwise approximated by homomorphisms with a precompact range. In this form, the property (which we call Kirchberg's property) makes sense for an arbitrary topological group. We establish the validity of the Kirchberg property for the isometry group $Iso(U)$ of the universal Urysohn metric space $U$ as a consequence of a stronger result: every representation of a residually finite group by isometries of $U$ can be pointwise approximated by representations with a finite range. This brings up the natural question of which other concrete infinite-dimensional groups satisfy the Kirchberg property.

math.RT↗

arXiv2000

A note on the precompactness of weakly almost periodic groups

An action of a group $G$ on a compact space $X$ is called weakly almost periodic if the orbit of every continuous function on $X$ is weakly relatively compact in $C(X)$. We observe that for a topological group $G$ the following are equivalent: (i) every continuous action of $G$ on a compact space is weakly almost periodic; (ii) $G$ is precompact. For monothetic groups the result was previously obtained by Akin and Glasner, while for locally compact groups it has been known for a long time.

math.GN↗

arXiv1999

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability of a unitary representation $π$ of a group $G$ (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_π,G)$, where $\s_\H$ is the unit sphere the Hilbert space of representation. We prove that $π$ is amenable if and only if either $π$ contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_π$ has a $G$-fixed point. Equivalently, the latter means that the $G$-space $(\s_π,G)$ has the concentration property: every finite cover of the sphere $\s_π$ contains a set $A$ such that for every $\e>0$ the $\e$-neighbourhoods of the translations of $A$ by finitely many elements of $G$ always intersect. As a corollary, amenability of $π$ is equivalent to the existence of a $G$-invariant mean on the uniformly continuous bounded functions on $\s_π$. As another corollary, a locally compact group $G$ is amenable if and only if for every strongly continuous unitary representation of $G$ in an infinite-dimensional Hilbert space $\mathcal H$ the system $(\s_\H,G)$ has the property of concentration.

math.FA↗

arXiv1994

Douady's conjecture on Banach analytic spaces

We show that, as conjectured by Adrien Douady back in 1972, every complete metric space is homeomorphic (moreover, isometric) to the locus of zeros of an analytic map between two Banach spaces. As a corollary, a paracompact topological space admits the structure of a Banach analytic space if and only if it is metrizable with a complete metric.

funct-an↗

arXiv1994

Regular Lie groups and a theorem of Lie-Palais

In 1984 Milnor had shown how to deduce the Lie-Palais theorem on integration of infinitesimal actions of finite-dimensional Lie algebras on compact manifolds from general theory of regular Lie groups modelled on locally convex spaces. We show how, in the case of effective action, one can eliminate from Milnor's argument the abstract Lie-Cartan theorem, making the deduction rather elementary. A machinery employed in the proof provides a partial solution to a problem examined in 1972 by van Est and Świerczkowski.

funct-an↗

arXiv1993

Operator spaces and residually finite-dimensional $C^\ast$-algebras

For every operator space $X$ the $C^\ast$-algebra containing it in a universal way is residually finite-dimensional (that is, has a separating family of finite-dimensional representations). In particular, the free $C^\ast$-algebra on any normed space so is. This is an extension of an earlier result by Goodearl and Menal, and our short proof is based on a criterion due to Exel and Loring.

funct-an↗

arXiv1992

An analytic structure emerging in presence of infinitely many odd coordinates

This is a contribution to the program of featuring even geometry as a ``collective effect in infinite-dimensional odd geometry,'' as suggested by Manin. We show that the (Gel'fand) spectrum of the locally convex nonstandard hull (in the sense of Luxemburg) of a grassmannian algebra with infinitely many odd generators contains a nontrivial analytic part.

funct-an↗

arXiv1992

Universal arrows to forgetful functors from categories of topological algebra

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and much more. An accent is put on relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and others may hopefully become amenable to methods of this theory.

funct-an↗

arXiv1992

Nonstandard hulls of Banach-Lie groups and algebras

We propose a new construction of Banach-Lie groups and algebras relying on nonstandard analysis. A major standard application is the Local Theorem which to certain extent reduces the problem of associating a Lie group to a given banach-Lie algebra to a similar problem for finitely generated Lie subalgebras. We discuss possible applications, e.g., to gauge theories.

funct-an↗