Unitary connections on Bratteli diagrams

TL;DR

This paper extends Ocneanu's theory of graph connections to a 2-categorical framework involving tracial Bratteli diagrams and unitary connections, embedding it into hyperfinite von Neumann algebras.

Contribution

It introduces a new 2-category structure for connections on Bratteli diagrams and embeds it into hyperfinite von Neumann algebras, generalizing subfactor theory results.

Findings

Defined a 2-category with tracial Bratteli diagrams and unitary connections

Embedded this 2-category into the 2-category of hyperfinite von Neumann algebras

Generalized fundamental subfactor theory results to a 2-categorical setting

Abstract

In this paper, we extend Ocneanu's theory of connections on graphs to define a 2-category whose 0-cells are tracial Bratteli diagrams, and whose 1-cells are generalizations of unitary connections. We show that this 2-category admits an embedding into the 2-category of hyperfinite von Neumann algebras, generalizing fundamental results from subfactor theory to a 2-categorical setting.