Type III subfactors and planar algebras

Shelly, Claire 2012. Type III subfactors and planar algebras. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

In this thesis we investigate the theory of planar algebras for type III subfactors. We show directly how to associate a planar algebra to a type III subfactor using endomorphisms and intertwiners. We begin by describing how to define a type III version of the Temperley-Lieb planar algebra before giving a general definition of a type III planar algebra. We define a presenting map using endomorphisms and intertwiners and prove that this defines a type III subfactor planar algebra. We show that the definition of a type III subfactor planar algebra may be extended by removing the sphericality condition. We also investigate the reverse implication, and show that if we start with a type III subfactor planar algebra we can produce a type III subfactor using techniques from Guionnet-Jones-Shlyakhtenko and free probability. In the final chapter we investigate the type III version of A2 planar algebras. We extend the results of Chapter 3 to the A2 setting, defining a type III string algebra for SU(3) ADE graphs and relating this to planar algebras. We also discuss further work here relating to A2 planar algebras.

Item Type:	Thesis (PhD)
Status:	Unpublished
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Funders:	EU-NCG network in Non-Commutative Geometry MRTN-CT-2006-031962
Date of First Compliant Deposit:	30 March 2016
Last Modified:	19 Mar 2016 23:15
URI:	https://orca.cardiff.ac.uk/id/eprint/44565

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