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Isaac Goldbring

Publications and source records attributed to Isaac Goldbring.

At least 19 recordsLinked to original sources
arXiv2026

Ultrapowers of spectral subspaces

We prove, for any W$^*$-probability space $(M,\varphi)$ where $M$ is a type $\mathrm{III}_1$ factor, any nontrivial, proper closed $F\subseteq \mathbb{R}$, and any nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, that the ultrapower $M(\sigma^\varphi,F)^{\mathcal{U}}$ of the spectral subspace $M(\sigma^\varphi,F)$ is a proper subset of the spectral subspace $M^{\mathcal{U}}(\sigma^{\varphi^{\mathcal{U}}},F)$. We discuss the model-theoretic implications of this result.

math.OA↗
arXiv2026

Model theory and Connes' bicentralizer problem

We make a series of model-theoretic contributions to Connes' bicentralizer problem, one of the most prominent open problems in the theory of von Neumann algebras. Our work builds on the recent result of Houdayer and Marrakchi who show that, for separable diffuse W$^*$-probability spaces, having trivial bicentralizer is equivalent to being selfless, that is, having the first factor inclusion into the free product be an existential embedding. We first show that the class of selfless $W^*$-probability spaces is $\forall\exists$-axiomatizable. We then extend the Houdayer-Marrakchi equivalence to all diffuse W$^*$-probability spaces, removing the separability hypothesis. Combining these results, we show that for any axiomatizable class of diffuse $W^*$-probability spaces, those with trivial bicentralizer form an $\forall\exists$-axiomatizable class; in particular, the class of type $\mathrm{III}_1$ factors with trivial bicentralizer is $\forall\exists$-axiomatizable. We give concrete axioms for this class using totally bounded variants of Haagerup's characterization of the bicentralizer, which we develop here and believe to be of independent interest. We also introduce the notion of pseudoperiodic $\mathrm{III}_1$ factors and show that any such factor has trivial bicentralizer. In the final section, we prove that the bicentralizer problem has a positive solution if and only if the bicentralizer functor is a zeroset relative to the theory of $\mathrm{III}_1$ factors. We use this result to give an equivalent formulation of the bicentralizer problem in terms of a uniformity condition on Haagerup's Dixmier-type characterization of the bicentralizer.

math.OA↗
arXiv2026

Computable $K$-theory for C*-algebras II: AF algebras

We continue the study of the effective content of $K$-theory for C*-algebras, with a focus on AF algebras. We show that from a c.e. presentation of an AF algebra it is possible to compute a representation of the algebra as an inductive limit of finite-dimensional algebras. Using this, and an analogous result for dimension groups, we show that the computable $K_0$ functor provides a computable equivalence of categories between c.e. presentations of AF algebras and c.e. presentations of unital (scaled) dimension groups, giving an effective version of Elliott's classification theorem. We use our results to determine the complexity of the index set and isomorphism problems for various classes of AF algebras.

math.OA↗
arXiv2025

A nonstandard approach to the direct integral version of the Spectral Theorem

We use nonstandard methods to prove the direct integral version of the Spectral Theorem for Unbounded Self-adjoint Operators. Our proof avoids the standard reduction to the case of bounded normal operators via the Cayley transform and, as such, works uniformly for both real and complex Hilbert spaces. Our method also yields a new nonstandard proof of the spectral measure version of the Spectral Theorem.

math.SP↗
arXiv2025

General real-valued theories with the Schr\"oder-Bernstein property are stable

We show that every general theory \`a la Keisler with the Schr\"oder-Bernstein property is stable. This generalizes the corresponding result from classical logic due to John Goodrick. Our proof uses the classical result (generalized to the case that the instability is witnessed by an infinitary formula) together with a discretization technique introduced by Keisler and the third-named author. We speculate on how our techniques could be adapted to show that every continuous theory with the Schr\"oder-Bernstein property is stable.

math.LO↗
arXiv2025

On definability of C*-tensor norms

We initiate the study of definability (in the model-theoretic sense) of C*-tensor norms. We show that neither the minimal nor maximal tensor norms are definable uniformly over all C*-algebras. The proof in the case of the minimal tensor product norm uses a deep theorem of Kirchberg characterizing exactness in terms of tensor products with matrix ultraproducts while the case of maximal tensor products uses Pisier's recent characterization of the lifting property in terms of maximal tensor products and ultraproducts. We also study the question of when one of these tensor products can be definable in a particular C*-algebra. We establish some negative results along these lines for particular C*-algebras and when the definability condition is strengthened to be computable and of a restricted quantifier-complexity; these results use the quantum complexity results MIP$^*$=RE and MIP$^{co}=$coRE. As a byproduct of our arguments, we answer a question of Fritz, Netzer, and Thom by showing that the norm on $C^*(\mathbb{F}_n\times \mathbb{F}_n)$ is not computable for any $n\in \{2,3,\ldots,\infty\}$.

math.OA↗
arXiv2025

Model theory of Hilbert spaces expanded by normal operators

We study expansions of Hilbert spaces with a bounded normal operator $T$. We axiomatize this theory in a natural language and identify all of its completions. We prove the definability of the adjoint $T^*$ and prove quantifier elimination for every completion after adding $T^*$ to the language. We identify types with measures on the spectrum of the operator and show that the logic topology on the type space corresponds to the weak*-topology on the space of measures. We also give a precise formula for the metric on the space of $1$-types. We prove all completions are stable and characterize the stability spectrum of the theory in terms of the spectrum of the operator. We also show all completions, regardless of their spectrum, are $\omega$-stable up to perturbations.

math.LO↗
arXiv2025

Existentially closed measure-preserving actions of approximately treeable groups

Given a countable group $\Gamma$, letting $\mathcal{K}_\Gamma$ denote the class of {\pmp} actions of $\Gamma$, we study the question of when the model companion of $\mathcal{K}_\Gamma$ exists. Berenstein, Henson, and Ibarluc\'ia showed that the model companion of $\mathcal{K}_\Gamma$ exists when $\Gamma$ is a nonabelian free group on a countable number of generators. We significantly generalize their result by showing that the model companion of $\cal K_\Gamma$ exists whenever $\Gamma$ is an approximately treeable group. The class of approximately treeable groups contain the class of treeable groups as well as the class of universally free groups, that is, the class of groups with the same universal theory as nonabelian free groups. We prove this result using an open mapping characterization of when the model companion exists; moreover, this open mapping characterization provides concrete, ergodic-theoretic axioms for the model companion when it exists. We show how to simplify these axioms in the case of treeable groups, providing an alternate axiomatization for the model companion in the case of the free group, which was first axiomatized by Berenstein, Henson, and Ibarluc\'ia using techniques from model-theoretic stability theory. Along the way, we prove a purely ergodic-theoretic result of independent interest, namely that finitely generated universally free groups (also known as limit groups) have Kechris' property MD. We also show that for groups with Kechris' EMD property, the profinite completion action is existentially closed, and for groups without property (T), the generic existentially closed action is weakly mixing, generalizing results of Berenstein, Henson, and Ibarluc\'ia for the case of nonabelian free groups.

math.LO↗
arXiv2025

Computable presentations of randomizations

We initiate the effective metric structure theory of Keisler randomizations. We show that a classical countable structure $\mathcal{M}$ has a decidable presentation if and only if its Borel randomization $\mathcal{M}^{[0,1)}$ has a computable presentation for which the constant functions are uniformly computable points. We determine a sufficient condition for which the uniform computability of the constant functions can be dropped. We show that when $\mathcal{M}$ is effectively $\omega$-categorical, then $\mathcal{M}^{[0,1)}$ is computably categorical, that is, has a unique computable presentation up to computable isomorphism. A special case of this result is that the unique separable atomless probability algebra is computably categorical. Finally, we show that all randomizations admit effective quantifier elimination.

math.LO↗
arXiv2025

The Schr\"oder-Bernstein property for operators on Hilbert spaces

We establish that the complete theory of a Hilbert space equipped with a normal operator has the Schr\"oder-Bernstein property. This answers a question of Argoty, Berenstein, and the first-named author. We also prove an analogous statement for unbounded self-adjoint operators.

math.LO↗
arXiv2025

Totally Bounded Elements in W*-probability Spaces

We introduce the notion of a totally ($K$-) bounded element of a W*-probability space $(M, \varphi)$ and, borrowing ideas of Kadison, give an intrinsic characterization of the $^*$-subalgebra $M_{tb}$ of totally bounded elements. Namely, we show that $M_{tb}$ is the unique strongly dense $^*$-subalgebra $M_0$ of totally bounded elements of $M$ for which the collection of totally $1$-bounded elements of $M_0$ is complete with respect to the $\|\cdot\|_\varphi^\#$-norm and for which $M_0$ is closed under all operators $h_a(\log(\Delta))$ for $a \in \mathbb{N}$, where $\Delta$ is the modular operator and $h_a(t):=1/\cosh(t-a)$ (see Theorem 4.3). As an application, we combine this characterization with Rieffel and Van Daele's bounded approach to modular theory to arrive at a new language and axiomatization of W*-probability spaces as metric structures. Previous work of Dabrowski had axiomatized W*-probability spaces using a smeared version of multiplication, but the subalgebra $M_{tb}$ allows us to give an axiomatization in terms of the original algebra operations. Finally, we prove the (non-)axiomatizability of several classes of W*-probability spaces.

math.OA↗
arXiv2025

Computable $K$-theory for $\mathrm{C}^*$-algebras: UHF algebras

We initiate the study of the effective content of $K$-theory for $\mathrm{C}^*$-algebras. We prove that there are computable functors which associate, to a computably enumerable presentation of a $\mathrm{C}^*$-algebra $\boldA$, computably enumerable presentations of the abelian groups $K_0(\boldA)$ and $K_1(\boldA)$. When $\boldA$ is stably finite, we show that the positive cone of $K_0(\boldA)$ is computably enumerable. We strengthen the results in the case that $\boldA$ is a UHF algebra by showing that the aforementioned presentation of $K_0(\boldA)$ is actually computable. In the UHF case, we also show that $\boldA$ has a computable presentation precisely when $K_0(\boldA)$ has a computable presentation, which in turn is equivalent to the supernatural number of $\boldA$ being lower semicomputable; we give an example that shows that this latter equivalence cannot be improved to requiring that the supernatural number of $\boldA$ is computable. Finally, we prove that every UHF algebra is computably categorical.

math.LO↗
arXiv2024

Effective subfactor theory

We characterize when a subfactor $N\subseteq M$ is oracle computable relative to a presentation of the ambient factor $M$ in terms of computability of the Jones basic construction, in terms of computable Pismner-Popa bases, and in terms of computability of the conditional expectation map. We illustrate our main theorem with some examples.

math.LO↗
arXiv2024

Computably strongly self-absorbing C*-algebras

We introduce the notion of a computably strongly self-absorbing C*-algebra and show that the following C*-algebras are computably strongly self-absorbing: the Cuntz algebras $\mathcal{O}_2$ and $\mathcal{O}_\infty$, the UHF algebra $M_{\mathfrak{n}}(\mathbb{C})$ and the tensor product $M_{\mathfrak{n}}(\mathbb{C})\otimes \mathcal{O}_\infty$, where $\mathfrak{n}$ is a supernatural number of infinite type with computably enumerable support, and the Jiang-Su algebra $\mathcal{Z}$. In connection with the last example, we show that $\mathcal{Z}$ has a computable presentation. The results above are a special instance of a computable version of the standard approximate intertwining argument due to Elliott.

math.LO↗
arXiv2024

Undecidability and incompleteness in quantum information theory and operator algebras

We survey a number of incompleteness results in operator algebras stemming from the recent undecidability result in quantum complexity theory known as $\operatorname{MIP}^*=\operatorname{RE}$, the most prominent of which is the G\"odelian refutation of the Connes Embedding Problem. We also discuss the very recent use of $\operatorname{MIP}^*=\operatorname{RE}$ in refuting the Aldous-Lyons conjecture in probability theory.

math.LO↗
arXiv2024

Back-and-forth equivalent group von Neumann algebras

We prove that if $G$ and $H$ are $\alpha$-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal $\alpha \geq \omega$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also $\alpha$-back-and-forth equivalent. In particular, if $G$ and $H$ are $\omega$-back-and-forth-equivalent groups, then $L(G)$ and $L(H)$ are elementarily equivalent; this is known to fail under the weaker hypothesis that $G$ and $H$ are merely elementarily equivalent. We extend this result to crossed product von Neumann algebras associated to Bernoulli actions of back-and-forth equivalent groups.

math.LO↗
arXiv2024

Computable Gelfand Duality

We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative $C^*$ algebras.

math.LO↗