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Totally Bounded Elements in W*-probability Spaces

Jananan Arulseelan, Isaac Goldbring, Bradd Hart, Thomas Sinclair

TL;DR

We address the model-theoretic treatment of σ-finite W*-probability spaces by introducing totally $K$-bounded elements as the natural sorts and proving that the strongly dense subalgebra of totally bounded elements $M_{ ext{tb}}$ is uniquely characterized by completeness of $S_1(M_0)$ in $\|\cdot\|_{\\varphi}^{\\#}$ and closure under $h_a(\\log \Delta)$. This intrinsic Kadison-inspired characterization enables a concrete axiomatization using ordinary algebraic operations via the bounded-operator approach to Tomita–Takesaki theory, and yields a metric-structure framework for W*-probability spaces compatible with modular dynamics. The paper also establishes a robust connection between Ocneanu ultraproducts and model-theoretic ultraproducts, and provides both negative and positive results on the axiomatizability of natural classes, thereby integrating modular theory with computable, first-order-style descriptions of W*-probability spaces. Together, these results lay the groundwork for a concrete, modular-language approach to W*-probability spaces and their ultraproducts, with implications for both theory and applications in noncommutative probability and operator-algebra model theory.

Abstract

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 $h_a(\log(Δ))$ 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.

Totally Bounded Elements in W*-probability Spaces

TL;DR

We address the model-theoretic treatment of σ-finite W*-probability spaces by introducing totally -bounded elements as the natural sorts and proving that the strongly dense subalgebra of totally bounded elements is uniquely characterized by completeness of in and closure under . This intrinsic Kadison-inspired characterization enables a concrete axiomatization using ordinary algebraic operations via the bounded-operator approach to Tomita–Takesaki theory, and yields a metric-structure framework for W*-probability spaces compatible with modular dynamics. The paper also establishes a robust connection between Ocneanu ultraproducts and model-theoretic ultraproducts, and provides both negative and positive results on the axiomatizability of natural classes, thereby integrating modular theory with computable, first-order-style descriptions of W*-probability spaces. Together, these results lay the groundwork for a concrete, modular-language approach to W*-probability spaces and their ultraproducts, with implications for both theory and applications in noncommutative probability and operator-algebra model theory.

Abstract

We introduce the notion of a totally (-) bounded element of a W*-probability space and, borrowing ideas of Kadison, give an intrinsic characterization of the -subalgebra of totally bounded elements. Namely, we show that is the unique strongly dense -subalgebra of totally bounded elements of for which the collection of totally -bounded elements of is complete with respect to the -norm and for which is closed under all operators for , where is the modular operator and (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 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.
Paper Structure (8 sections, 39 theorems, 45 equations)

This paper contains 8 sections, 39 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $\mathrm{W}^*$-probability spaces
  3. Tomita-Takesaki theory according to Rieffel and Van Daele
  4. Characterization of $M_{\operatorname{tb}}$
  5. Axiomatizating $\mathrm{W}^*$-probability spaces
  6. Connection with the Ocneanu ultraproduct
  7. Expanding the language by the modular automorphism group
  8. Axiomatizable and local classes

Key Result

Lemma 2.1

For $v\in H$, we have that $v\in M'\omega$ if and only if the map extends to a bounded operator on $H$. In particular, $a\in M$ is right-bounded if and only if $a\omega\in M'\omega$, in which case $\pi'(a)$ is the unique $z\in M'$ such that $a\omega=z\omega$.

Theorems & Definitions (70)

  • Lemma 2.1
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Proposition \ref{['phidense']}
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 60 more