We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a… (More)

Let N ⊆ M be II1 factors with [M : N ] < ∞. There is a “standard invariant” for N ⊆ M , which we shall describe using the planar algebra formalism of [19]. The vector spaces Pk of N −N invariant… (More)

The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. In this sense it is opposite in… (More)

We prove that finiteness of the index of the intersection of a finite set of finite index subalgebras in a von Neumann algebra (with small centre) is equivalent to the finite dimensionality of the… (More)

We give a diagrammatic description of Popa’s symmetric enveloping algebras associated to planar algebra subfactors. As an application we construct a natural family of derivations on these factors,… (More)

A subfactor is an inclusion N ⊂ M of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action M ⊂M , and subfactors can be thought of as fixed… (More)

A link is a finite family of disjoint, smooth, oriented or unoriented, closed curves in R or equivalently S. A knot is a link with one component. The Jones polynomial VL(t) is a Laurent polynomial in… (More)

Many very different themes could be used for a talk such as this one, but I have chosen von Neumann algebras because they are what led me into this circle of ideas. Thus the presentation will be… (More)