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Many people I talk to lament the nonexistence of a coherent source for learning the theory of subfactors.

Could someone suggest a nice (ordered) list of books/papers to work through to obtain a suitable background in this theory, assuming the audience is comprised of mathematicians familiar with the basics of von Neumann algebra theory?

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3 Answers 3

I also think a good starting point is Jones and Sunders. If you are interested in type III subfactors, the literature seems to me a bit rare, but there are some lecture notes:

Kosaki - "Type III factors and index theory"

In algebraic quantum field theory one studies "nets" which associates to certain space time regions (under certain conditions) a type III factor. Subfactors play an important role. There is e.g. the paper:

Longo, Rehren - "Nets of Subfactors"

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Ah, those lecture notes by Kosaki look interesting, didn't know they exist. Thanks! –  Pieter Naaijkens Nov 11 '10 at 12:04
    
Thanks, Marcel! –  Jon Bannon Nov 11 '10 at 17:46

So it really depends on why you want to learn about subfactors. I'll try and give different reading lists based on different motivation.

The basics of $II_1$-subfactors:

If you're familiar with $II_1$-factors, then "Introduction to subfactors" (MR1473221) is a good place to start. Of course, Jones' original "Index for subfactors" (MR696688) is an enjoyable read as well. A more advanced and comprehensive treatment is Evans and Kawahigashi's "Quantum symmetries on operator algebras" (MR1642584).

Classification:

When mathematicians encounter a family of mathematical objects, we feel the need to classify them. Subfactors were first (and still!) classified by their principal graphs, which is the principal part of the Bratteli diagram of the standard invariant, or tower of relative commutants (see JS, EK, or Bisch's "Bimodules and higher relative commutants" MR1424954). The first classifications (index $\leq 4$) were completed by Jones, Ocneanu (MR996454) (very hard to find this source), and Popa (MR1278111). Popa showed that "amenable" subfactors of the hyperfinite $II_1$-factor (the index is the norm squared of the principal graph) are completely classified by their principal graphs.

Axiomatization:

There are several axiomatizations for the standard invariant of a subfactor: Ocneanu's paragroups (see EK), Popa's $\lambda$-lattices (MR1334479), and Jones' planar algebras (see section below on planar algebras). These three different axiomatizations play together nicely, and it's good to have an overview on what's really going on here. Unfortunately, there is no good unified source for this. Yet. However, you should think of it this way:

The standard invariant (or "representation theory") of a finite index $II_1$-subfactor $N\subset M$ is a unitary $2$-category with $2$ $0$-morphisms called $N$ and $M$, $1$-morphisms given by various bimodule summands of the basic constructions of $N\subset M$, and $2$-morphisms given by various intertwiner spaces. This is what Ocneanu calls a "paragroup" because it resembles the tensor category of representations of a finite group. This $2$-category is unitary, has nice duals, and satisfies Frobenius reciprocity, and other cool stuff as well. In particular, we can draw planar diagrams to represent different $2$-morphisms in the spaces. So this $2$-category has the structure of a planar algebra. For this category theory stuff, see Mueger's "From subfactors to categories and topology I" (MR1966524). In many cases, we can recover the special $2$-category from connections on a bipartite graph or on a commuting square (see EK, JS, Popa).

Reconstruction:

Popa proved that one can start with he standard invariant of a subfactor and reconstruct a subfactor with the same standard invariant (MR1334479). In the "strongly amenable" case (a bit more technical than "amenable"), you get a subfactor of the hyperfinite $II_1$-factor. You can also do the reconstruction completely planar algebraically. This result is due to Guionnet-Jones-Shlyakhtenko (arXiv:0712.2904), and a really easy version to understand was given by Jones-Shlyakhtenko-Walker (arXiv:0807.4146) (Kodiyalam and Sunder also have a version of this).

Planar Algebras:

If you want to know what a planar algebra is, see Peters' construction of the Haagerup subfactor planar algebra (arXiv:0902.1294), or Morrison-Peters-Snyder "Skein-theory for the $D_{2n}$ planar algebras" (MR2559686). If you want to know how a subfactor actually gives a planar algebra, see the first section of Jones-Penneys (arXiv:1007.3173), which relies on some proofs in Jones' "Planar Algebras I" (arXiv:math/9909027).

Examples:

A great class of examples is the Bisch-Haagerup subfactors (MR1386923) which are just slightly harder than group-subgroup examples (see JS). Some of the most important examples rely on the above reconstruction theorems. For example, there are the exotic Haagerup and Asaeda-Haagerup subfactors (MR1686551) and the composite Fuss-Catalan subfactors (MR1437496).

I'll keep updating this post. Right now I have office hours. Sections to come include:

Type III, Recent results

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Excellent, Dave! Thanks!!!! –  Jon Bannon Oct 27 '10 at 0:59

Maybe the best choice today is the book by Jones and Sunder, "Introduction to subfactors.", at least the first few chapters to get a grip of the basics. After this it depends on your interests in which direction you could continue. I don't know enough about the field to give any specific references.

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This is probably a good start. I have this book and have read it. I guess the question really should be where to go from there? There is a lot more to subfactor theory than is contained in that book, if I'm not mistaken. –  Jon Bannon Oct 26 '10 at 17:14

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