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Isaac Goldbring

Publications and source records attributed to Isaac Goldbring.

At least 55 records · Page 3Linked to original sources
arXiv2021

The Connes Embedding Problem: A guided tour

The Connes Embedding Problem (CEP) is a problem in the theory of tracial von Neumann algebras and asks whether or not every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite II$_1$ factor. The CEP has had interactions with a wide variety of areas of mathematics, including C*-algebra theory, geometric group theory, free probability, and noncommutative real algebraic geometry (to name a few). After remaining open for over 40 years, a negative solution was recently obtained as a corollary of a landmark result in quantum complexity theory known as $\operatorname{MIP}^*=\operatorname{RE}$. In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from $\operatorname{MIP}^*=\operatorname{RE}$. In fact, we outline two such proofs, one following the "traditional" route that goes via Kirchberg's QWEP problem in C*-algebra theory and Tsirelson's problem in quantum information theory and a second that uses basic ideas from logic.

math.OA↗
arXiv2021

The Universal Theory Of The Hyperfinite II$_1$ Factor Is Not Computable

We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg's QWEP Conjecture and Tsirelson's Problem.+

math.LO↗
arXiv2021

On the complexity of the theory of a computably presented metric structure

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in $[0,1]$, we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $ϕ^\mathcal{M} \leq r$, and the open diagram, which encapsulates strict inequalities of the form $ϕ^\mathcal{M} < r$. We show that the closed and open $Σ_N$ diagrams are $Π^0_{N+1}$ and $Σ_N$ respectively, and that the closed and open $Π_N$ diagrams are $Π^0_N$ and $Σ^0_{N + 1}$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.

math.LO↗
arXiv2021

Everettian mechanics with hyperfinitely many worlds

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett's limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. This approach also provides a more general framework in which to consider no-collapse formulations of quantum mechanics more generally.

quant-ph↗
arXiv2021

Non-embeddable II$_1$ factors resembling the hyperfinite II$_1$ factor

We consider various statements that characterize the hyperfinite II$_1$ factors amongst embeddable II$_1$ factors in the non-embeddable situation. In particular, we show that "generically" a II$_1$ factor has the Jung property (which states that every embedding of itself into its ultrapower is unitarily conjugate to the diagonal embedding) if and only if it is self-tracially stable (which says that every such embedding has an approximate lifting). We prove that the enforceable factor, should it exist, has these equivalent properties. Our techniques are model-theoretic in nature. We also show how these techniques can be used to give new proofs that the hyperfinite II$_1$ factor has the aforementioned properties.

math.OA↗
arXiv2020

Properties expressible in small fragments of the theory of the hyperfinite II_1 factor

We show that any II$_1$ factor that has the same 4-quantifier theory as the hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\mathcal{R}$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $\mathcal{R}$ is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $\mathcal{R}$ has the same 3-quantifier theory as an infinitely generic embeddable factor.

math.OA↗
arXiv2020

Operator algebras with hyperarithmetic theory

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(Γ)$ for $Γ$ a finitely generated group with solvable word problem, $C^*(Γ)$ for $Γ$ a finitely presented group, $C^*_λ(Γ)$ for $Γ$ a finitely generated group with solvable word problem, $C(2^ω)$, and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problem has an affirmative answer. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. C$^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. C$^*$-algebras).

math.OA↗
arXiv2020

Factorial relative commutants and the generalized Jung property for II$_1$ factors

We introduce the notion of a generalized Jung factor: a II$_1$ factor $M$ for which any two embeddings of $M$ into its ultrapower $M^{\mathcal U}$ are equivalent by an automorphism of $M^{\mathcal U}$. We show that $\mathcal R$ is not the unique generalized Jung factor but is the unique $\mathcal R^{\mathcal U}$-embeddable generalized Jung factor. We use model-theoretic techniques to obtain these results. Integral to the techniques used is the result that if $M$ is elementarily equivalent to $\mathcal R$, then any elementary embedding of $M$ into $\mathcal R^{\mathcal U}$ has factorial relative commutant. This answers a long-standing question of Popa for an uncountable family of II$_1$ factors. We also provide new examples and results about the notion of super McDuffness, which is a strengthening of the McDuff property for II$_1$ factors.

math.OA↗
arXiv2020

On Popa's factorial commutant embedding problem

An open question of Sorin Popa asks whether or not every $R^{\mathcal{U}}$-embeddable factor admits an embedding into $R^{\mathcal{U}}$ with factorial relative commutant. We show that there is a locally universal McDuff II$_1$ factor $M$ such that every property (T) factor admits an embedding into $M^{\mathcal{U}}$ with factorial relative commutant. We also discuss how our strategy could be used to settle Popa's question for property (T) factors if a certain open question in the model theory of operator algebras has a positive solution.

math.OA↗
arXiv2020

Oracle computability of conditional expectations onto subfactors

We initiate the effective study of conditional expectations onto subfactors. Our main result is that if $M$ is an existentially closed II$_1$ factor with a w-spectral gap subfactor $N$, then the conditional expectation function onto $N$ can be computed from a (Turing) oracle that computes a presentation of $M$, the inclusion of $N$ into $M$, and a spectral gap function for the pair $(M,N)$.

math.OA↗
arXiv2019

Correspondences, Ultraproducts and Model Theory

We study correspondences of tracial von Neumann algebras from the model-theoretic point of view. We introduce and study an ultraproduct of correspondences and use this ultraproduct to prove, for a fixed pair of tracial von Neumann algebras M and N, that the class of M-N correspondences forms an elementary class. We prove that the corresponding theory is classifiable, all of its completions are stable, that these completions have quantifier elimination in an appropriate language, and that one of these completions is in fact the model companion. We also show that the class of triples (M, H, N), where M and N are tracial von Neumann algebras and H is an M-N correspondence, form an elementary class. As an application of our framework, we show that a II_1 factor M has property (T) precisely when the set of central vectors form a definable set relative to the theory of M-M correspondences. We then use our approach to give a simpler proof that the class of structures (M, Phi), where M is a sigma-finite von Neumann algebra and Phi is a faithful normal state, forms an elementary class. Finally, we initiate the study of a family of Connes-type ultraproducts on C*-algebras.

math.LO↗
arXiv2019

The Urysohn sphere is pseudofinite

We show that the Urysohn sphere is pseudofinite. As a consequence, we derive an approximate $0$-$1$ law for finite metric spaces of diameter at most $1$.

math.LO↗
arXiv2019

Continuous Sentences Preserved Under Reduced Products

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

math.LO↗
arXiv2019

On the nonexistence of Følner sets

We show that there is $n\in \mathbf N$, a finite system $Σ(\vec x,\vec y)$ of equations and inequations having a solution in some group, where $\vec x$ has length $n$, and $ε>0$ such that: for any group $G$ and any $\vec a\in G^n$, if the system $Σ(\vec a,\vec y)$ has a solution in $G$, then there is no $(\vec a,ε)$-Fø lner set in $G$. The proof uses ideas from model-theoretic forcing together with the observation that no amenable group can be existentially closed. Along the way, we also observe that no existentially closed group can be exact, have the Haagerup property, or have property (T). Finally, we show that, for $n$ large enough and for $ε$ small enough, the existence of $(F,ε)$-Fø lner sets, where $F$ has size at most $n$, cannot be expressed in a first-order way uniformly in all groups.

math.GR↗
arXiv2018

A nonstandard take on central sets

We present the basic theory of central subsets of semigroups from the nonstandard perspective. A key feature of this perspective is the replacement of the algebra of ultrafilters with the algebra of elements of iterated hyperextensions, a technique first employed by Mauro Di Nasso.

math.CO↗
arXiv2018

Independence in randomizations

The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of $T$. We show that if $T$ has the exchange property and $\operatorname{acl}=\operatorname{dcl}$, then $T^R$ has a strict independence relation in the home sort, and hence is real rosy. In particular, if $T$ is o-minimal, then $T^R$ is real rosy.

math.LO↗