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

TL;DR

This paper establishes a noncommutative Minkowski integral inequality for von Neumann algebras, providing a criterion for categorifying fusion rings via graph inclusion conditions.

Contribution

It introduces a noncommutative Minkowski inequality and derives a novel unitary categorification criterion for fusion rings from graph inclusion properties.

Findings

Proves a noncommutative Minkowski integral inequality for von Neumann algebras.

Provides a necessary condition for graphs to be realized as inclusion graphs of matrix algebra quadruples.

Derives a categorification criterion for fusion rings based on graph properties.

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.