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Hyperlinear and sofic groups: a brief guide
 Bull. Symbolic Logic
"... 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 re ..."
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Cited by 19 (1 self)
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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.
The isometry group of the Urysohn space as a Lévy Group
, 2005
"... 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 m ..."
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Cited by 8 (3 self)
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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. We propose a reformulation of Connes’ Embedding Conjecture as an approximationtype statement about the unitary group U(ℓ²), and show that in this form the conjecture makes sense also for Iso(U).
Remarks on automorphisms of ultrapowers of II1 factors
"... Abstract. In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II1 factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ0locally i ..."
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Cited by 4 (3 self)
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Abstract. In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II1 factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is ℵ0locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *homomorphisms from a separable nuclear C*algebra into an ultrapower of a II1 factor, equality of the induced traces implies unitary equivalence. All statements are proved using operator algebraic techniques, but in the last section of the paper we indicate how the underlying principle is related to theorems of Henson’s positive bounded logic. 1.
Representations of residually finite groups by isometries of the Urysohn space
 Journal of the Ramanujan Mathematical Society
"... Abstract. As a consequence of Kirchberg’s work, Connes ’ Embedding Conjecture is equivalent to the property that every homomorphism of the group F ∞ × F ∞ into the unitary group U(ℓ 2) with the strong topology is pointwise approximated by homomorphisms with a precompact range. In this form, the pro ..."
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Cited by 2 (2 self)
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Abstract. As a consequence of Kirchberg’s work, Connes ’ Embedding Conjecture is equivalent to the property that every homomorphism of the group F ∞ × F ∞ into the unitary group U(ℓ 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 infinitedimensional groups satisfy the Kirchberg property. 1.
Model theory of operator algebras III: Elementary equivalence and II1 factors, preprint
, 1111
"... Abstract. We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II1 factors and their ultrapowers. Among other things, we show that for any II1 factor M, there are continuum many nonisomorphic separable II1 factors that have an ultrapower isomorphic ..."
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Cited by 1 (1 self)
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Abstract. We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II1 factors and their ultrapowers. Among other things, we show that for any II1 factor M, there are continuum many nonisomorphic separable II1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man’s resolution of the Connes Embedding Problem: there exists a separable II1 factor such that all II1 factors embed into one of its ultrapowers. 1.
On finite approximations of topological algebraic systems
, 2006
"... We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept s ..."
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We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from K using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula ϕ holds in A if and only if all precise enough approximations of ϕ hold in all precise enough approximations of A. We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be noncommutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings. 1
DOUBLEDUAL TYPES OVER THE BANACH SPACE C(K)
, 2005
"... Let K be a compact Hausdorff space and C(K) the Banach space of all realvalued continuous functions on K, with the supnorm. Types over C(K) (in the sense of Krivine and Maurey) can be uniquely represented by pairs (ℓ,u) of bounded realvalued functions on K, whereℓ is lower semicontinuous, u is up ..."
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Let K be a compact Hausdorff space and C(K) the Banach space of all realvalued continuous functions on K, with the supnorm. Types over C(K) (in the sense of Krivine and Maurey) can be uniquely represented by pairs (ℓ,u) of bounded realvalued functions on K, whereℓ is lower semicontinuous, u is upper semicontinuous, ℓ ≤ u, andℓ(x) = u(x) for all isolated points x of K. A condition that characterizes the pairs (ℓ,u) that represent doubledual types over C(K)isgiven. 1. Statement of the main theorem The concept of type over a Banach space E was first introduced by Krivine and Maurey [7] in the context of separable Banach spaces. The reader is referred to Garling’s monograph [4] for more details. We consider general, not necessarily separable Banach spaces. Let E be a Banach space. For every x ∈ E, we define a function τx: E → R by letting τx(y) = ‖x + y ‖ for all y ∈ E. Definition 1.1. A function τ: E → R is a type over E if τ is in the closure (with respect to the topology of pointwise convergence) of the set {τx: x ∈ E}. The definition given here is equivalent to the definition given in [1]. That is, τ is a type over E if and only if there exists an ultrafilter � over an infinite index set λ and a bounded family of elements (xα)α∈λ in E such that τ(y) = limα∈ � ‖xα + y ‖ for all y ∈ E.Thereader is referred to [5] for more details regarding the choice of the ultrafilter. Throughout, we let K be a compact Hausdorff topological space. The topology on K is denoted by Ω. Weletℓ∞(K) denote the Banach lattice of bounded realvalued functions on K equipped with the supnorm. For f,g ∈ ℓ∞(K), the lattice ordering is defined pointwise. An sc pair (semicontinuous pair) is a pair of functions (ℓ,u)fromℓ∞(K)suchthatℓ is lower semicontinuous (lsc), u is upper semicontinuous (usc), ℓ ≤ u, andℓ(x) = u(x) for all isolated points x ∈ K. The Banach space of continuous realvalued functions on K with supnorm is denoted by C(K). The constant function with value 1 is denoted by 1.
© Hindawi Publishing Corp. DOUBLEDUAL nTYPES OVER BANACH SPACES NOT CONTAINING ℓ1
, 2002
"... Let E be a Banach space. The concept of ntype over E is introduced here, generalizing the concept of type over E introduced by Krivine and Maurey. Let E ′ ′ be the second dual of E and fix g ′′ 1,...,g′ ′ n ∈ E′ ′. The function τ: E × Rn → R, defined by letting τ(x,a1,...,an) = ‖x + ∑n i=1 aig ′′ i ..."
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Let E be a Banach space. The concept of ntype over E is introduced here, generalizing the concept of type over E introduced by Krivine and Maurey. Let E ′ ′ be the second dual of E and fix g ′′ 1,...,g′ ′ n ∈ E′ ′. The function τ: E × Rn → R, defined by letting τ(x,a1,...,an) = ‖x + ∑n i=1 aig ′′ i ‖ for all x ∈ E and all a1,...,an ∈ R,definesanntype over E. Types that can be represented in this way are called doubledual ntypes; we say that (g ′′ 1,...,g′ ′ n) ∈ (E′′) n realizes τ. LetEbe a (not necessarily separable) Banach space that does not contain ℓ1. We study the set of elements of (E′′) n that realize a given doubledual ntype over E. Weshow that the set of realizations of this ntype is convex. This generalizes a result of Haydon and Maurey who showed that the set of realizations of a given 1type over a separable Banach space E is convex. The proof makes use of Henson’s language for normed space structures and uses ideas from mathematical logic, most notably the LöwenheimSkolem theorem. 2000 Mathematics Subject Classification: 46B04, 46B20. 1. Introduction. We