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Noncommutative Minkowski integral inequality and a unitary categorification criterion for fusion rings

Junhwi Lim

TL;DR

The paper develops a noncommutative Minkowski-type integral inequality for commuting squares of tracial von Neumann algebras, supported by a generalized Hölder inequality and a central variant. It then demonstrates how this inequality yields a practical criterion for testing whether graphs can realize a commuting square and, consequently, whether a given fusion or based ring admits a unitary categorification. Applying the criterion to large datasets of fusion rings, the authors exclude a substantial fraction and provide insights into which rings remain candidates under existing Drinfeld-type and related criteria. Overall, the work builds a bridge between operator-algebraic inequalities and the realizability of unitary fusion and based categories, offering concrete tools for classification and exclusion in categorical representation theory.

Abstract

We prove a noncommutative analogue of Minkowski's integral inequality for commuting squares of tracial von Neumann algebras. The inequality implies a necessary condition for a quadruple of graphs to be realized as inclusion graphs of a commuting square of multi-matrix algebras. As a corollary, we obtain a unitary categorification criterion for based rings, in particular, fusion rings.

Noncommutative Minkowski integral inequality and a unitary categorification criterion for fusion rings

TL;DR

The paper develops a noncommutative Minkowski-type integral inequality for commuting squares of tracial von Neumann algebras, supported by a generalized Hölder inequality and a central variant. It then demonstrates how this inequality yields a practical criterion for testing whether graphs can realize a commuting square and, consequently, whether a given fusion or based ring admits a unitary categorification. Applying the criterion to large datasets of fusion rings, the authors exclude a substantial fraction and provide insights into which rings remain candidates under existing Drinfeld-type and related criteria. Overall, the work builds a bridge between operator-algebraic inequalities and the realizability of unitary fusion and based categories, offering concrete tools for classification and exclusion in categorical representation theory.

Abstract

We prove a noncommutative analogue of Minkowski's integral inequality for commuting squares of tracial von Neumann algebras. The inequality implies a necessary condition for a quadruple of graphs to be realized as inclusion graphs of a commuting square of multi-matrix algebras. As a corollary, we obtain a unitary categorification criterion for based rings, in particular, fusion rings.
Paper Structure (8 sections, 13 theorems, 38 equations)

This paper contains 8 sections, 13 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Noncommutative Minkowski's integral inequality
  4. Reduced Hölder inequality
  5. Noncommutative Minkowski's integral inequality
  6. Applications
  7. Construction of commuting squares
  8. Unitary categorification of fusion rings

Key Result

Theorem A

Let $(\mathcal{M},\mathop{\mathrm{tr}}\nolimits)$ be a tracial von Neumann algebra, $\mathcal{N}\subset \mathcal{L},\mathcal{K}\subset \mathcal{M}$ be von Neumann subalgebras and $x\in \mathcal{M}$ be a positive element satisfying the following: Then for $1\le p<\infty$ we have

Theorems & Definitions (28)

  • Theorem A
  • Theorem B
  • Corollary C
  • Theorem 2.1: NCLp
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: Reduced Hölder inequality
  • proof
  • Proposition 2.5
  • proof
  • ...and 18 more