Notes of Claus Koestler’s Cambridge lecture 26-01-2017

An elementary approach to unitary representations of Thompson’s group {F}

{F=\langle X_n \,|\,X_k X_n=X_{n+1}X_k,\,0\leq k<n\rangle}. Unknown wether amenable or not. Does not contain free subgroups.

1. Sources for unitary representations of {F}

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 {{\mathbb Z}^2}) plus {\{0,0\}} (left regular representation).

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

  1. {F} is non-alenable.
  2. {C^*_r(F)} has a unique normalized trace.

Not so productive.

1.2. Subfactor approach

Theorem 3 (Jones 2014) TFAE Any subfactor yields a unitary representation of {F}.

1.3. Probabilistic approach

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

1.4. Graphical picture

for the monoid {F^+} (Belk 2004).

Represent generators {X_0} and {X_1} as diagrams: collection of edges joining an {n+1}-point set to an {n}-point set, with two edges joining at 0 (resp. at 1). Such diagram can be composed. Get a category whose objects are finite sets {([n])_{n\in{\mathbb N}}} and morphisms are finite binary forests.

In the semicosimplicial category {\Delta_S}, same objects, morphisms are increasing functions. They satisfy again the relations of {F}.

There is a covariant functor from {\Delta_S} 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 {D}. Let {U} and {V} be unitaries {D\rightarrow D\oplus D}. Inserting them in partial shifts {S_k}, get unitaries {U_k}.

Theorem 5 The unitaries {T_k:=U_{k+1}S_{k+1}} satisfy Thompson {F}‘s relations.

We call this the standard form representation of {F}.

Theorem 6 Let {\pi:F\rightarrow\mathcal{H}} be a unitary representation such that

  1. No fixed vectors.
  2. {\mathcal{H}} is generated by fixed vectors of generators {X_n}.

Then {\pi} has standard form up to modifications of {S_k}‘s.

2.1. Outlook

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

Study the {C^*} and von Neumann algebras of {F}.

2.2. Question

What if you embed {F} 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 http://www.math.ens.fr/metric2011/
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