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
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.
Does the V. Jones’ example fit into this framework ?
Study the and von Neumann algebras of .
What if you embed into diffeos of the circle and compose with unitary representations coming from conformal field theory ?