## Notes of Stefan Vaes 1st Cambridge lecture 17-01-2017

Representation theory and cohomology for standard invariants

We study pairs ${N\subset M}$ of ${II_1}$ factors, with ${dim_N M}$ finite.

There are a bunch of discrete invariants.

• Standard invariants
• Rigid ${C^*}$ algebras
• Planar algebras (Jones).

How can these invariants act on factors ? This is Popa’s theory.

We shall introduce representations on Hilbert spaces, and define ${L^2}$ Betti numbers.

1. Overview: standard invariants

1.1. Jones tower

With a pair ${N\subset M}$, there comes a whole tower ${N\subset M\subset M_1\subset\cdots}$, with equal indices. ${M_1=}$. ${B_{ij}}$ is a grid of multimatrix algebras with ${e_n\in B_{ij}}$ if ${i). Popa calls such a structure a ${\lambda}$-lattice.

1.2. Alternative view

View ${M}$ as an ${N-M}$-bimodule. By tensor products over ${N}$, one gets more modules ${M\otimes_N M\otimes_N M\cdots}$. Then, as an ${M-M}$-bimodule, ${M_1=M\otimes M}$. Also, ${End_{N-N}(M)\simeq N'\cap M_1}$. We get a 2-category (of ${N-M}$-bimodules) with generator ${_N M_M}$.

1.3. Planar algebras

This is a diagrammatic way to write intertwiners ${End_{M-M}(M\otimes_N M\otimes\cdots)}$.

1.4. Popa’s classification

Theorem 1 (Popa 1992) The standard invariant is a complete invariant of the pair when ${N\simeq M}$ is the hyperfinite ${II_1}$ factor and the standard invariant is amenable.

This suggests to define amenability, property (T), Haagerup property,… for the standard invariants. This was done by Popa. In 2014, Popa and I gave an intrinsic definition for these notions fir ${\lambda}$-lattices and tensor categories. Very soon, Neshveyev-Yamashita defined unitary representations (the Drinfeld center). Immediately after, Ghosh-C. Jones showed that unitary representations coincide with ordinary representations of the tube algebra.

2. The symmetric-enveloping inclusion

Now we start again with a more synthetic view.

Start with ${N\subset M}$. Let ${T=M\otimes M^{op}}$. Let ${S}$ be the unique ${II_1}$ factor generated by ${T}$ and a projection and such that

• ${}$ are the toric construction for ${N\otimes 1\subset M\otimes 1}$.
• Idem for ${N^{op}}$.

This automatically contains the entire Jones tower and also the Jones tunnel (a tower contained in ${M}$). Then

$\displaystyle \begin{array}{rcl} M_m=(1\otimes M_{-n}^{op})' \cap S. \end{array}$

Longo and Rehren, Morsodo have an alternative construction for ${S}$. Consider the category ${\mathcal{C}}$ of all finite index ${M}$-bimodules appearing in ${M^{\otimes_N^k}}$. Every irredcible ${\alpha\in Irr(\mathcal{C})}$ is realised by ${_M(H_{\alpha})_M}$. Define

$\displaystyle \begin{array}{rcl} S_0 = \bigoplus_{\alpha\in Irr(\mathcal{C})}(H_\alpha\otimes \bar H_\alpha). \end{array}$

This has an obvious product.

Theorem 2 There is an involution ${*}$ and a positive functional ${\tau}$,

$\displaystyle \begin{array}{rcl} \tau(\xi_1\otimes\bar \xi_2)=\begin{cases} 0 & \text{ if }\alpha\not=\text{trivial}, \\ \text{trace on }M\otimes M^{op} & \text{otherwise}. \end{cases} \end{array}$

Hence we get a pair of von Neumann algebras ${T\subset S}$. ${\tau}$ is a trace when ${N\subset M}$ is extremal (meaning that all bimodules in ${\mathcal{C}}$ have equal left and right dimension.

3. Quasiregular inclusions

${T\subset S}$ is called quasiregular if ${S}$ belongs to the closure of the span of finite index subbimodules.

Example 1. The symmetric-enveloping inclusion just described is.

Example 2. Crossed products ${S=T\times \Gamma}$.

The more general approach (by Popa, Shlyakhtenko and myself) goes as follows.

A “unitary respresentation” for ${T\subset S}$ is a Hilbert ${S}$-bimodule ${H}$ such that as a ${T}$-bimodule ${_T H_T}$, we have a direct sum of ${T}$-bimodules in a given category ${\mathcal{C}}$.

A “positive definite function” is a completely positive ${T}$-linear map ${\phi:S\rightarrow S}$. In the symmetric-enveloping example, such maps ${\phi}$ are scalar on each ${H_\alpha\otimes \bar H_\alpha}$, therefore it boils down to a ${{\mathbb C}}$-valude fonction on ${Irr(\mathcal{C})}$. There in an obvious DNS construction that passes from positive definite functions to unitary representations.

4. Plan

In lecture 2, I will attach to ${T\subset S}$ an ${*}$-algebra (like ${{\mathbb C}\Gamma}$).

In lecture 3, I will do cohomology, ${L^2}$-Betti numbers (like ${{\mathbb C}\Gamma\subset L\Gamma}$).

In lecture 4, I will give examples and computations.