The embedding theorem for finite depth subfactor planar algebras

TL;DR

This paper establishes a canonical construction of a relative commutant planar algebra from finite von Neumann algebra inclusions and demonstrates its isomorphism to bipartite graph planar algebras, linking subfactor theory with graph theory.

Contribution

It introduces a canonical relative commutant planar algebra for finite von Neumann algebra inclusions and proves its isomorphism to bipartite graph planar algebras in finite-dimensional cases.

Findings

Canonical relative commutant planar algebra defined from von Neumann algebra inclusions

Isomorphism with bipartite graph planar algebra in finite-dimensional cases

Finite depth subfactor planar algebra embeds into bipartite graph planar algebra

Abstract

We define a canonical relative commutant planar algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional C*-algebras with the Markov trace, we show this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.