Unshaded planar algebras and their associated II_1 factors

TL;DR

This paper introduces unshaded planar algebras and constructs associated II_1 factors, demonstrating that these factors contain a specific maximal abelian subalgebra, expanding the framework of subfactor theory.

Contribution

It defines unshaded planar algebras and constructs associated II_1 factors, extending Guionnet et al.'s work to a broader class of planar algebras.

Findings

M_P is a II_1 factor

M_P contains a maximal abelian subalgebra called the cup subalgebra

The construction generalizes previous subfactor algebra frameworks

Abstract

Guionnet et al. gave a construction of a II_1 factor associated to a subfactor planar algebra. In this paper we define an unshaded planar algebra. To any unshaded planar algebra P we associate a finite von Neumann algebra M_P. We prove that M_P is a II_1 factor that contains a generic maximal abelian subalgebra called the cup subalgebra.