arXiv subjects

Jananan Arulseelan

Publications and source records attributed to Jananan Arulseelan.

9 recordsLinked to original sources

arXiv2026

Every $\mathrm{AW}^*$-Algebra is Normal

Using set-theoretic methods, we prove that every $\mathrm{AW}^*$-algebra is normal, resolving a question of Wright that has stood open for 46 years. This was previously known in the case of $\mathrm{AW}^*$-factors, by work of Sait\^o and Wright. We also show that if there is a model $V$ of $\mathrm{ZFC}$ with an $\mathrm{AW}^*$-algebra that fails to be monotone complete, then in some forcing extension of $V$ there is an $\mathrm{AW}^*$-factor that fails to be monotone complete, implying that any $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-factors are monotone complete yields a $\mathrm{ZFC}$ proof that all $\mathrm{AW}^*$-algebras are monotone complete. Both results build on transfer principles for Boolean-valued factor representations originally developed by Ozawa. This work was assisted by the Danus LLM orchestration system.

math.OA↗

arXiv2026

Pointwise Mean Value Theorems in Constructive Mathematics

We answer some questions regarding the mean value theorem and related results in constructive mathematics. The answers to these questions reveal interesting properties of the Mean Value Theorem, Law of Bounded Change, and Constancy Principle. We see that, in contrast to the Intermediate Value Theorem whose approximate analogue was shown to hold constructively by Frank, the natural approximate versions of these theorems fail to hold in neutral constructive mathematics. Our proof of this makes use of the existence of a topos in which the real numbers are in bijection with the naturals, which was shown by Bauer and the second author. Using Booij's notion of locators, we show that the aforementioned approximate analogues do hold in the presence of a small amount of countable choice, and also under suitable locator lifting hypotheses. We also show that an even weaker approximate analogue of the Mean Value Theorem holds in neutral constructive mathematics.

math.LO↗

arXiv2026

A Note on Pseudofinite W*-Probability Spaces

We introduce pseudofinite W*-probability spaces. These are W*-probability spaces that are elementarily equivalent to Ocneanu ultraproducts of finite-dimensional von Neumann algebras equipped with arbitrary faithful normal states. We are particularly interested in the case where these finite-dimensional von Neumann algebras are full matrix algebras: the pseudofinite factors. We show that these are indeed factors. We see as a consequence that pseudofinite factors are never of type $\mathrm{III}_0$. Mimicking the construction of the Powers factors, we give explicit families of examples of matrix algebra ultraproducts that are $\mathrm{III}_\lambda$ factors for $\lambda \in (0,1]$. We show that these examples share their universal theories with the corresponding Powers factor and thus have uncomputable universal theories. Finally, we show that pseudofinite factors are full. This generalizes a theorem of Farah-Hart-Sherman which shows that pseudofinite tracial factors do not have property $\Gamma$. It has the consequence that hyperfinite factors of type $\mathrm{III}$ (the Powers factors) are never pseudofinite. Our proofs combine operator algebraic insights with routine continuous logic syntactic arguments: using \L os' theorem to prove that certain sentences which are true for all matrix algebras are inherited by their ultraproducts.

math.OA↗

arXiv2025

Model Theory of General von Neumann Algebras II: Group Actions and Crossed Products

Expanding on previous work of the author, we initiate the model theoretic study of W$^*$-dynamical systems. We axiomatize continuous weight-preserving group actions of $G$ on von Neumann algebras for $G$ a given locally compact Hausdorff group. Since our axiomatization is of continuous actions, the ultraproduct is defined so that the ultraproduct action of $G$ is also continuous. Building on a theorem of Tomatsu, we show that continuous ultraproducts commute with crossed products. Finally, we prove a suite of results about computability of the aforementioned axiomatizations and of presentations of crossed products. In particular, we show how the crossed product construction is a useful tool for producing computable presentations, giving special attention to the group measure space construction of Murray and von Neumann. Thereby, we establish interesting connections to computable dynamics and computable measure theory.

math.OA↗

arXiv2025

The Universal Theory of Locally Universal Tracial von Neumann Algebras is not Computable

Building on Lin's breakthrough MIP$^{co}$ = coRE and an encoding of non-local games as universal sentences in the language of tracial von Neumann algebras, we show that locally universal tracial von Neumann algebras have undecidable universal theories. This implies that no such algebra admits a computable presentation. Our results also provide, for the first time, explicit examples of separable II$_1$ factors without computable presentations, and in fact yield a broad family of them, including McDuff factors, factors without property Gamma, and property (T) factors. We also obtain analogous results for locally universal semifinite von Neumann algebras and tracial C*-algebras. The latter provides strong evidence for a negative solution to the Kirchberg Embedding Problem. We discuss how these are obstructions to approximation properties in the class of tracial and semifinite von Neumann algebras.

math.OA↗

arXiv2025

Model Theory of General von Neumann Algebras I: Generalized Ocneanu Ultraproducts

This paper collates, presents, and expands upon technology and results obtained as part of the author's PhD thesis. We generalize work done in the $\sigma$-finite setting by the author, Goldbring, Hart, and Sinclair by producing a language and axiomatization of full left Hilbert algebras. To improve the accessibility of using this axiomatization, we examine the metric structure ultraproduct associated to this axiomatization. In doing so, we generalize results of Ando-Haagerup and Masuda-Tomatsu. This examination leads us to multiple operator-algebraic characterizations of the ultraproduct which closely resemble known characterizations of the Ocneanu ultraproduct. One of these is closely related to the notion of continuous elements of an ultraproduct with respect to an action. In the spirit of results of the author, Goldbring, and Hart, we prove various undecidable universal theory results for von Neumann algebras with unbounded weights. Notably, we prove that the hyperfinite II$_\infty$ factor together with its canonical tracial weight has an undecidable theory. This result was expected by experts, but could not be made precise until now. We also study the strengthenings of the negative solution to CEP that this result implies.

math.OA↗

arXiv2025

Totally Bounded Elements in W*-probability Spaces

We introduce the notion of a totally ($K$-) bounded element of a W*-probability space $(M, φ)$ 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 $\|\cdot\|_φ^\#$-norm and for which $M_0$ is closed under all operators $h_a(\log(Δ))$ for $a \in \mathbb{N}$, where $Δ$ 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↗

arXiv2024

On the Elementary Equivalence of Group Factors

We use techniques of distortion systems, introduced by James Hanson to show that any two elementarily equivalent infinite discrete ICC groups give rise to elementarily equivalent group von Neumann algebras. This answers a question raised by Goldbring-Pi (and attributed there to Koichi Oyakawa), in the negative. In a special case, this resolves the model theoretic free factor isomorphism problem.

math.OA↗

arXiv2023

The undecidability of having the QWEP

We show that neither the class of C*-algebras with Kirchberg's QWEP property nor the class of W*-probability spaces with the QWEP property are effectively axiomatizable (in the appropriate languages). The latter result follows from a more general result, namely that the hyperfinite III$_1$ factor does not have a computable universal theory in the language of W*-probability spaces. We also prove that the Powers' factors $\mathcal{R}_λ$, for $0<λ<1$, when equipped with their canonical Powers' states, do not have computable universal theory.

math.OA↗