** An elementary approach to unitary representations of Thompson’s group **

. Unknown wether amenable or not. Does not contain free subgroups.

**1. Sources for unitary representations of **

** 1.1. Traditional approach **

Characters!

**Theorem 1 (Gohm-Kostler, Dudko-Madzrodski)** * Extreme points of the set of characters and in one to one correspondance with points of the 2-torus (characters of the abelianisation ) plus (left regular representation). *

**Theorem 2 (Gohm-Kostler, Dudko-Madzrodski)** * TFAE *

*
*
- is non-alenable.
- has a unique normalized trace.

* *

Not so productive.

** 1.2. Subfactor approach **

**Theorem 3 (Jones 2014)** * TFAE Any subfactor yields a unitary representation of . *

** 1.3. Probabilistic approach **

**Theorem 4 (Gohm, Evans, Bhat, Wills, C. Jones)** * Every non-commutative stationary Markov chain yields a unitary representation of . *

** 1.4. Graphical picture **

for the monoid (Belk 2004).

Represent generators and as diagrams: collection of edges joining an -point set to an -point set, with two edges joining at 0 (resp. at 1). Such diagram can be composed. Get a category whose objects are finite sets and morphisms are finite binary forests.

In the semicosimplicial category , same objects, morphisms are increasing functions. They satisfy again the relations of .

There is a covariant functor from to the category of NonCommutativePS. With Evans et de Finetti, we related coface identities to spreadability and non-commutative Bernoulli shift.

**2. From Markov chains to representations **

Start with construction on infinite sets, kind of limit of the previous one: partial shifts.

Pass to Hilbert spaces: partial shifts on sequences of vectors in a fixed Hilbert space . Let and be unitaries . Inserting them in partial shifts , get unitaries .

**Theorem 5** * The unitaries satisfy Thompson ‘s relations. *

We call this the standard form representation of .

**Theorem 6** * Let be a unitary representation such that *

*
*
- No fixed vectors.
- is generated by fixed vectors of generators .

* Then has standard form up to modifications of ‘s. *

** 2.1. Outlook **

Does the V. Jones’ example fit into this framework ?

Study the and von Neumann algebras of .

** 2.2. Question **

What if you embed into diffeos of the circle and compose with unitary representations coming from conformal field theory ?

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metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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