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=<M,e_0>}. {B_{ij}} is a grid of multimatrix algebras with {e_n\in B_{ij}} if {i<n<j}). 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

  • {<M\otimes 1,e>} 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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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