arXiv subjects

Itaï Ben Yaacov

Publications and source records attributed to Itaï Ben Yaacov.

51 records · Page 3Linked to original sources

arXiv2009

Randomizations of models as metric structures

The notion of a randomization of a first order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new structure whose elements are random elements of the original first order structure. In this paper we treat randomizations as continuous structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the earlier results show that the randomization of a complete first order theory is a complete theory in continuous logic that admits elimination of quantifiers and has a natural set of axioms. We show that the randomization operation preserves the properties of being omega-categorical, omega-stable, and stable.

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arXiv2009

Reflexive representability and stable metrics

It is well-known that a topological group can be represented as a group of isometries of a reflexive Banach space if and only if its topology is induced by weakly almost periodic functions (see \cite{Shtern:CompactSemitopologicalSemigroups}, \cite{Megrelishvili:OperatorTopologies} and \cite{Megrelishvili:TopologicalTransformations}). We show that for a metrisable group this is equivalent to the property that its metric is uniformly equivalent to a stable metric in the sense of Krivine and Maurey (see \cite{Krivine-Maurey:EspacesDeBanachStables}). This result is used to give a partial negative answer to a problem of Megrelishvili.

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arXiv2009

Definability of groups in $\aleph_0$-stable metric structures

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate}

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arXiv2009

Model-theoretic independence in the Banach lattices $L_p(μ)$

We study model-theoretic stability and independence in Banach lattices of the form $L_p(X,U,μ)$, where $1 \leq p < \infty$. We characterize non-dividing using concepts from analysis and show that canonical bases exist as tuples of real elements.

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arXiv2009

On perturbations of Hilbert spaces and probability algebras with a generic automorphism

We prove that $IHS_A$, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is $\aleph_0$-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, $APr_A$, the theory of atomless probability algebras equipped with a generic automorphism is $\aleph_0$-stable up to perturbation. However, not allowing perturbation it is not even superstable.

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arXiv2009

Continuous first order logic for unbounded metric structures

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach (which has been the common approach so far to Banach space structures in continuous logic), as well as of applying in situations where the unit ball approach does not apply (i.e., when the unit ball is not a definable set). We also introduce the process of single point \emph{emboundment} (closely related to the topological single point compactification), allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together with results from \cite{BenYaacov:Perturbations} regarding perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson.

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arXiv2009

On perturbations of continuous structures

We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $\aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations (where the notion of perturbation is part of the data). As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations.

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arXiv2009

A proof of completeness for continuous first-order logic

The primary purpose of this article is to show that a certain natural set of axioms yields a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an \emph{approximated} form of strong completeness, whereby $Σ\vDashφ$ (if and) only if $Σ\vdashφ\dotminus 2^{-n}$ for all $n<ω$. This approximated form of strong completeness asserts that if $Σ\vDashφ$, then proofs from $Σ$, being finite, can provide arbitrary better approximations of the truth of $φ$.

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arXiv2009

Modular functionals and perturbations of Nakano spaces

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin \cite{Poitevin:PhD}. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and that complete theories of atomless Nakano spaces are model complete. We also show that up to arbitrarily small perturbations of the exponent Nakano spaces are $\aleph_0$-categorical and $\aleph_0$-stable. In particular they are stable.

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arXiv2009

On supersimplicity and lovely pairs of cats

We prove that the definition of supersimplicity in metric structures from \cite{pezz:Morley} is equivalent to an \textit{a priori} stronger variant. This stronger variant is then used to prove that if $T$ is a supersimple Hausdorff cat then so is its theory of lovely pairs.

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arXiv2009

Lovely pairs of models: the non first order case

We prove that for every simple theory $T$ (or even simple thick compact abstract theory) there is a (unique) compact abstract theory $T^\fP$ whose saturated models are the lovely pairs of $T$. Independence-theoretic results that were proved in \cite{ppv:pairs} when $T^\fP$ is a first order theory are proved for the general case: in particular $T^\fP$ is simple and we characterise independence.

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arXiv2008

Model theoretic forcing in analysis

We present a framework for model theoretic forcing in a non-first-order context, and present some applications of this framework to Banach space theory.

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arXiv2008

Topometric spaces and perturbations of metric structures

We develop the general theory of \emph{topometric spaces}, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric function. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop a theory of Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the \textit{ad hoc} development from \cite{BenYaacov-Usvyatsov:CFO}), as well as of global $\aleph_0$-stability. We conclude with a study of perturbation systems (see \cite{BenYaacov:Perturbations}) in the formalism of topometric spaces. In particular, we show how the abstract development applies to $\aleph_0$-stability up to perturbation.

math.LO↗

arXiv2008

Continuous and Random Vapnik-Chervonenkis Classes

We show that if $T$ is a dependent theory then so is its Keisler randomisation $T^R$. In order to do this we generalise the notion of a Vapnik-Chervonenkis class to families of $[0,1]$-valued functions (a \emph{continuous} Vapnik-Chervonenkis class), and we characterise families of functions having this property via the growth rate of the mean width of an associated family of convex compacts.

math.LO↗

arXiv2008

Continuous first order logic and local stability

We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.

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